Algebra
Real algebra questions asked by students, solved step by step Every question below was submitted by a real student and answered step by step.
There is no solution at all. Simplifying the right side gives |x| = -5, and an absolute value is a distance from zero, so it can never equal a negative number.
The only solution is x = -5/2. When an absolute value equals zero there is no case split, because zero is the only number whose distance from zero is zero.
The solutions are x = 0 and x = 10/3. Because |A| = 5 forces A = 5 or A = -5, this single absolute value equation becomes two ordinary linear equations.
The solutions are x = -1/4 and x = -5/4. Split |4x + 3| = 2 into the two cases 4x + 3 = 2 and 4x + 3 = -2, then solve each linear equation separately.
The solutions are x = 0 and x = 10. Split the equation into 5 - x = 5 and 5 - x = -5; the negative coefficient of x is what makes the second case give x = 10.
The solutions are x = 3.4 and x = -3.4. The minus sign inside the bars changes nothing, because a number and its opposite are the same distance from zero.
The solutions are x = 5 and x = 3/23. Pull the constants outside the bars, cross-multiply to 8|2x − 3| = 7|x + 3|, then square to get 23x² − 118x + 15 = 0.
The solutions are x = -1 and x = 2. Splitting at x = 0 and x = 1 gives three cases; the middle case collapses to 1 = 3 and therefore has no solution.
The answer is -1 <= x <= 2. A less-than-or-equal absolute value becomes one closed double inequality, -3 <= 2x - 1 <= 3, solved by adding 1 and halving.
The answer is -9 < x < 4. A less-than absolute value splits into one double inequality -13 < 2x + 5 < 13, which you solve by subtracting 5 and halving.
The answer is w <= 40/7 or w >= 44/7. Divide by 7 first, then a greater-than-or-equal absolute value splits into two outward rays instead of one interval.
The solution is w <= 3 or w >= 9. Divide by 7 to reach |w - 6| >= 3, then read that as: w sits at least 3 units away from 6, in either direction.
The solution is a < -1 or a > 1/3. With absolute values on both sides, squaring is reversible, turning the problem into the quadratic 3a^2 + 2a - 1 > 0.
The solution is m < -9 or m > 9. Isolate |3m|, divide by -2 and flip the sign to get |3m| > 27, then split the outward inequality into two branches.
The solution is all real numbers. After dividing by -2 the inequality reads |3m| > -24, and since an absolute value is never negative it always holds.
The answer is x <= (11 - sqrt 97)/4 or x >= 5, about x <= 0.288 or x >= 5. Breakpoints at 2, 3 and 4 give four cases; only the outer two have solutions.
The solution is 4/3 <= x <= 2. A variable right-hand side must first be forced non-negative, and then the sandwich -(x-1) <= 2x-3 <= x-1 is solved.
The solution is x < -2, x = 1, or x > 2. The numerator is never negative, so the sign follows the denominator, and x = 1 makes the whole fraction exactly zero.
The solution of (|x − 2| − x)/(x² − 4) ≤ 0 is (−2, 1] ∪ (2, +∞). Split the absolute value at x = 2, build a sign table, and see why x = 1 is in but ±2 are out.
The sum is 7 times the fifth root of 4. Because both terms already share the same index and radicand, only their coefficients 2 and 5 are added together.
The sum equals 37 times the square root of 2. Simplify root 50 into 5 root 2 and root 32 into 4 root 2, then add the resulting coefficients 25 and 12.
The sum is -x^3 + 8x^2 - 15x - 4 and its degree is 3. Add the coefficients of matching powers; the cubic terms do not cancel, so the degree stays 3.
The sum is −4x² − 11x + 13. See how to line up like terms by degree, why the lone −11x carries straight through, and how to check the result at x = 1.
The sum equals sqrt(10). Squaring gives 6 + 2sqrt(9-5) = 10, and since both radicals are positive the sum is the positive root, not -sqrt(10).
The answer is 6x^2 + yz. Dropping the brackets is safe because the groups are added, not subtracted, and the two z^2 terms cancel exactly against each other.
The sum is 2(x^2 - 8x + 32) / (x(x - 8)). Cross-multiplying onto the denominator x(x - 8) gives x^2 + (x - 8)^2, which expands and factors to 2(x^2 - 8x + 32).
The older brother is 25 3/5 years old. Let the younger age be x, write 8x + 8 = 3(x + 8), solve 5x = 16, then multiply back to get the older age.
Answer: the older brother is 128/5 = 25.6 years old and the younger is 16/5 = 3.2. Setting the younger age to x gives 8x + 8 = 3(x + 8), so 5x = 16.
The child is 18 and the father is 72. The equation 4x - 12 = 10(x - 12) gives x = 18, and twelve years ago the ages were 6 and 60, a ratio of exactly 10.
I am 20 and my mother is 60. Subtracting 10 from both ages gives 3x - 10 = 5(x - 10), so x = 20; ten years ago the ages were 10 and 50, a ratio of 5.
The ages are 88/3 and 50/3 years, about 29.33 and 16.67. Two conditions give two equations; substitution reduces them to 3D - 4 = 46.
From u_8 = u_1 + 7d we get 7d = 26 - 1/3 = 77/3, so d = 11/3. Substituting back gives u_8 = 1/3 + 7(11/3) = 78/3 = 26, confirming the fraction.
Since u_4 = 10 the second equation gives u_6 = 16. Because u_6 - u_4 = 2d, we get 2d = 6 and d = 3, with u_1 = 1 reconstructing the whole sequence.
From u_4 - u_2 = 2d we get d = 2 and u_1 = 1. Then u_15 = 1 + 14(2) = 29, so the fifteenth term is 29 and not the tempting 31.
The nth-term rule u_n = u_1 + (n - 1)d with u_1 = 3 and d = 2 gives u_5 = 3 + 4(2) = 11. Listing 3, 5, 7, 9, 11 confirms the fifth term is 11.
The solution is x <= -9/2, that is the interval (-infinity, -9/2]. Split the chain into two inequalities, watch every x^2 term cancel, and intersect the results.
log 1576 / log 2.44 = log base 2.44 of 1576 = 8.2541. The shared base cancels, and raising 2.44 to the 8.2541 power returns 1575.99, confirming the exponent.
Answer: only the last two are equal. Expression 1 simplifies to 0.015S - 0.955 and expressions 2 and 3 both to 0.0138462S - 0.789231; they agree only at S = 431/3.
There are 30 five-unit coins and 30 ten-unit coins. Writing the second count as 60 - x turns the two conditions into the single equation 5x + 10(60 - x) = 450.
There are 30 five-unit coins and 50 ten-unit coins. The single equation 5x + 10(80 - x) = 650 encodes both the count and the value condition at once.
The expression equals (2x + 2)² = 4(x + 1)². Spot the A² + 2A + 1 pattern with A = 2x + 1 instead of expanding, then factor the 2 out of the bracket.
The answer is x, not 1x or 0. Adding the coefficients gives 4 + 2 - 5 = 1, and a coefficient of 1 is written invisibly in front of the variable.
The answer is -9xy^2. All three terms carry the identical variable part xy^2, so only the coefficients combine: 6 - 3 - 12 works out to -9, and xy^2 rides along.
The answer is x. All three terms share the same variable, so only the coefficients add: 9 - 14 + 6 = 1, and a coefficient of 1 is never written out.
The simplified form is 2x² + 7x − 7. Group terms by degree: 3x² − x² = 2x², then 2x + 5x = 7x, and the constant −7 has nothing to pair with.
The simplified form is -x^5 - 15x^4 - 6x + 1. Only the two x^3 terms are like terms and they cancel, which drops the degree-3 term entirely.
Answer: x = -12/5 = -2.4. Group the three x-terms into -5x and the three constants into -12, then divide to get x = -12/5, written as a fraction.
-12xy + 9xy - 3xy = -6xy. All three terms carry the identical variable part xy, so only the coefficients are added: -12 + 9 - 3 = -6, and xy is left alone.
The answer is x ≈ 0.0011432627. All three terms are like terms, so the coefficients add to 87,000,451.12 before a single division finishes the problem.
a^3 + b^3 is always at least a^2b + ab^2, with equality only when a = b, because the difference factors as (a + b)(a - b)^2, a product of nonnegative pieces.
The answer is 3(x - 2/3)^2 + 2/3, minimum 2/3. The discriminant is -8, so the zeros are the complex pair (2 plus or minus i root 2)/3, not real numbers.
The answer is (x - 1)^2 + 2, so the minimum value is 2 at x = 1. The discriminant is -8, which is why this quadratic never factors over the real numbers.
Answer: no real solutions; x = -0.19717 +/- 0.13382i. Move -0.2 across, get 4.3957x^2 + 1.7334x + 0.2496 = 0, and read the negative discriminant -1.38399.
The solution is -3 < x <= -2, or (-3, -2]. Subtracting 4 from all three parts of the compound inequality isolates x in a single move without flipping signs.
The answer is 17/60 < x < 2/5. Multiply all three parts by 7, divide by 60, then subtract 1 - every step keeps the direction because both factors are positive.
The solution is 8 < y <= 16. Add 8 to all three parts, then divide all three by 3; because 3 is positive the inequality signs and their strict/non-strict status never change.
The solution is x > 7/5. The variable sits in both outer parts, so the chain must be split into two inequalities and their solution sets intersected.
The answer is -11/60 < x < -1/15. Both bounds come out negative because 60/7 is already about 8.57, so 1 + x has to be pulled below 1 to fit.
There is no solution. The inequality demands a number greater than 8 and less than 7 at once, and the algebra confirms it by producing reversed bounds.
The answer is -1/15 < x < 1/20. This is the window that straddles zero, because 60/7 is about 8.57 - already inside the target range at x = 0.
There is no solution. Once 4018 - 2734 is simplified to 1284, the chain demands 3818 ≤ 499x - 20y < 1284 and 1284 ≤ 33x - 78y < -2534, both impossible.
The answer is -1 <= x <= 3. Work on all three parts at once: add 1 everywhere to get -3 <= 3x <= 9, then divide by the positive 3 with no sign flip.
Answer: 15/2 < x <= 21/2, that is (7.5, 10.5]. Multiply all three parts by 12 to get 2 < 2x - 13 <= 8, add 13, then halve - the strict and closed ends are preserved.
A = x^5(x - 1)^5 / (x^2 + 1)^4. The power rule moves each coefficient into an exponent, and the negative coefficient sends its factor into the denominator.
Answer: 7 integers (-4, -3, -2, -1, 2, 3, 4). Factor both parts, note the denominator opens downward, and read the sign chart between the critical points -4, -1/5, 1 and 4.
There are 15 similarity classes. Over ℂ the Jordan form is a complete invariant, so the count is p(3)·p(4) = 3·5 = 15, one for each pair of integer partitions of the two multiplicities.
Answer: x = -59/3, about -19.667. Cube both sides to get 5 - 3x = 64; unlike a square root, a cube root needs no domain condition and creates no extraneous roots.
The real solution is x = −33. Cube both sides to clear the 1/3 exponent, solve 2x + 2 = −64, and see why an odd root never creates extraneous solutions.
The three real roots are x ≈ −7.4802e−8, 0.00106167 and 0.00814390. Scaling x by 10³ first turns the badly conditioned decimals into a well-behaved cubic.
Answer: x = 1 and x = (3 +/- sqrt(21))/2, about 3.7913 and -0.7913. The key step is spotting that 214221 = 101^2 x 21, so the square root simplifies exactly.
Answer: x = 1.3210837, the only real root. The discriminant -42,120,000 is negative, so Cardano gives one real root and two complex ones; no rational root exists.
Answer: x = -24.8617, -2.4502 and 2.2304. Vieta is the fastest check: the three roots must sum to -92.3/3.68 = -25.0815 and multiply to 500/3.68 = 135.87.
The solution set is (-inf, -2) U (1, 3). Find the three zeros, split the number line, and read the sign of the product on each piece with a sign chart.
The solution set is (−∞, −1) ∪ (−1, 2). Factoring out x³ + 1 gives (x+1)²(x²−x+1)(x−2) < 0, and only x − 2 can be negative, with x = −1 excluded.
Answer: (-inf, -1) union (2, 4). Three simple roots split the line into four intervals whose signs alternate, and a strict inequality excludes all three roots.
The polynomial is f(x) = 5(x + 3)(x - 6)(x - 10) = 5x³ - 65x² + 60x + 900. See how each zero becomes a factor and how the leading coefficient fixes the scale.
The only real root is x ≈ 0.00548128. Recognising 0.016875 = 3(0.005625) and 9.4921875e−5 = 3(0.005625)² collapses the cubic to a shifted perfect cube.
Answer: x = -6. The cubic is a near-miss for (x+4)^3, whose constant is 64; the extra 8 makes it (x+4)^3 = -8, so x + 4 must equal the real cube root -2.
The only real solution is x = 2. The cubic is (x − 1)³ − 1, so the equation becomes (x − 1)³ = 1 and the real cube root gives x − 1 = 1.
The roots are x = 1, 2 and 3. Testing x = 1 gives zero, synthetic division leaves x² − 5x + 6, and that factors as (x − 2)(x − 3).
The only real solution is x = 2 + ρ ≈ 3.3247180, where ρ is the plastic number. Factor the left side as (x−1)(x−2)(x−3), substitute y = x − 2, and solve y³ − y − 1 = 0.
The sum of the roots is 10. Subtract the linear pattern 4x - 3, factor the difference as a(x-1)(x-2)(x-3)(x-4), and apply Vieta - the constant a cancels.
The equation has no real solutions. The polynomial is exactly (x⁴ − x³ + x² − x + 1)² + 1, and a real square can never equal −1, so the minimum value is 1.
The answer is a = (root 5 - 1)c/4. The messy radical root(6 - 2root 5) denests to root 5 - 1, because 6 - 2root 5 is the perfect square of that difference.
The nested radical equals √6 − √5 ≈ 0.2137. Write 11 − √120 as 11 − 2√30 and find m + n = 11 with mn = 30, giving the perfect square (√6 − √5)².
The solution is the unbounded region with corner (2,4), bounded left by x = 2 and above by y = 6 - x. Solving each inequality for y makes the overlap easy to shade.
The answer is 8x. Instead of expanding both squares, use A² − B² = (A − B)(A + B) with A = 2x + 1 and B = 2x − 1 so the quadratic terms never appear.
4(x + 3)(x - 3) = 4(x^2 - 9) = 4x^2 - 36. The two brackets are a conjugate pair, so a^2 - b^2 collapses them in one step instead of a full FOIL expansion.
The answer is (2x + 3m)(2x − m)/48. Each term is already a perfect square, so a² − b² applies directly with a = x/3 and b = (2x/3 − m)/4, i.e. x/6 − m/4.
The answer is 8xy. Factoring as a difference of squares gives the factors 4y and 2x, so all the squared terms cancel before you ever expand anything.
Each 20-term block is a perfect cube, so the difference is A^3 - B^3 with A - B = (c - e)(y - b). Recognising the cube collapses 40 terms into three factors.
Answer: 0. Each fraction reduces to -(3a + 2), because the reversed denominators supply a factor of -1, so the difference cancels identically.
The difference quotient simplifies to 2x + h - 5. See how to expand f(x + h), subtract f(x), factor h out of the numerator and cancel it, step by step.
OP = 1. Substituting t = m^2 + n^2 turns the condition into t^2 + 4t - 5 = 0, whose roots are 1 and -5; only t = 1 survives because a sum of squares cannot be negative.
Answer: 2m - 7 + 31/(m + 5). Reorder into 2m^2 + 3m - 4, synthetic-divide with the root -5, and check that the numerator at m = -5 really equals the remainder 31.
Answer: x^2 - 2 with remainder 0. Long division clears in two steps, and grouping confirms it, since 2x^3+x^2-4x-2 equals x^2(2x+1) - 2(2x+1) exactly.
The quotient is x - 1/2 and the remainder is -1/2. Insert the missing 0x term, then divide leading terms twice; the non-monic 2x forces fractional quotient terms.
The quotient is x - 1/2 and the remainder is 5/2. Two rounds of dividing leading terms finish it, and the division identity confirms the fractional pieces.
Answer: 2x + 1, for x not equal to -3. Split the middle term using 6 and 1, factor by grouping into (2x+1)(x+3), then cancel the common binomial factor.
Answer: 3p + 7 + 5/(p - 1). Reorder into descending powers first, then synthetic-divide by the root 1; the remainder 5 equals the value of the numerator at p = 1.
The quotient is 3x³ + 6x² + 8x + 24 with remainder 47. See why the missing x³ term has to be written as 0x³ before dividing, and how to check with f(2) = 47.
Answer: 4p + 3 + 10/(p - 2). Reorder to 4p^2 - 5p + 4, divide synthetically by the root 2, and confirm the remainder by evaluating the numerator at p = 2.
The quotient is 2x^3 + x^2 - 4x - 2 with remainder 0, so x = 1 is a root. The bottom row of synthetic division gives the coefficients directly.
Answer: f^2 - 4f + 16, with no remainder. Write the missing f^2 and f terms as zeros, run synthetic division with -4, and recognise the sum-of-cubes factorisation.
Answer: j^2 + 4j + 16, remainder 0. Use coefficients 1, 0, 0, -64 with the root +4; this is the difference-of-cubes identity, so every sign in the quotient is positive.
The result is x^2 - x + 1 - 1/(x + 1). Long division on x^3 + 0x^2 + 0x + 0 gives quotient x^2 - x + 1 and remainder -1, which matches the remainder theorem.
The quotient is x² − 2x + 3 with remainder 0, so x − 1 is a factor. Follow all three divide-multiply-subtract rounds and the factor-theorem check.
The quotient is x + 2 with remainder 0. Learn why dividing by a quadratic gives a degree-1 quotient and only takes two rounds, with a factoring cross-check.
Answer: x + 5, with remainder 0. Long division cancels exactly, or factor by grouping: x^4+5x^3-3x-15 = (x+5)(x^3-3).
The quotient is x/3 - 5 and the remainder is 50. Dividing x^2 by 3x gives the fractional lead term x/3, after which the second step clears completely.
The quotient is x³ + 2x² + 17x + 20 with remainder −80x − 148. Reorder both polynomials by degree, insert zero placeholders, then run four divide-multiply-subtract rounds.
Answer: A = 5. Dividing 2x^3 - x^2 + 2x + 5 by x + 1 gives the quotient 2x^2 - 3x + 5, so the third (constant) quotient entry labelled A is 5.
The domain of f(x) = 1/(8/(x − 5) − 2) is (−∞, 5) ∪ (5, 9) ∪ (9, ∞). A complex fraction hides two restrictions: the inner denominator and the outer one.
The domain is (-infinity, infinity). Both f(x) = 2 - x^2 and g(x) = x^2 + 4x - 60 are polynomials defined for every real number, so their difference is too.
a = b is the correct conclusion. Combining b^2 = 4ac with 2b = a + 4c yields (a - 4c)^2 = 0, so a = 4c, and substituting back gives b = 4c as well.
The only root is x = −0.1. Equal squares force 5x + 7 = ±(5x − 6); the plus case gives the impossible 7 = −6, and the minus case gives 10x = −1.
The line is y = x/3 - 25/3. Two negatives in the slope give the positive 1/3, and the fractional intercept should be kept exact rather than rounded.
The line is y = 2x + 13. Subtracting a negative x-coordinate gives the run 3, and the steep slope 2 pushes the intercept well above both given points.
Every real number is a solution. Because 1 - x and x - 1 are opposites, their squares are equal, so the two sides are literally the same expression.
The solution is x = 11. Expanding both sides makes the x^2 terms equal (both 1/2 x^2), so they cancel and a linear equation in x is left behind.
The solution is x = 44. The x^2/12 term appears on both sides after expanding (x+3)^2, so it cancels and only a linear equation in twelfths and twentieths is left.
The value is 13/2. Adding the two cube terms first gives 20x^3 + 6x + 1, so only one power of 1/2 has to be computed.
The value is 62. Expanding first collapses the expression to 20x + 61, so the awkward fraction x = 1/20 never has to be squared.
The value is 400. Both 9x² and 4y² are squares, and 12xy is exactly twice their roots multiplied, so the expression collapses to (3x − 2y)² = 20².
The value is about -65.63. Keep ln(90.1) = 4.500920 to six places: rounding it early to 4.5017 shifts the answer to -65.85, an error of more than 0.22.
Answer: f(4) = 9. First check which condition x = 4 satisfies: it lies in 3 <= x <= 10, so the middle rule x^2 - 7 applies, giving 16 - 7 = 9.
The value is 0. Substituting m = 3 and n = 2 gives 3 - 18/6 = 3 - 3, and cancelling the n first shows the expression is really m - m^2/3, which is zero at m = 3.
Answer: 50. Expand the sigma notation into 3 + 8 + 15 + 24, or use the closed forms for the sum of squares minus the count of terms - both give 50.
The value is 44. Substitute first, clear the parentheses and the exponent before multiplying, and only then subtract: 64 - 5(4) = 64 - 20 = 44.
The value is 25. The expression is the perfect square (x − 4y)², so the given relation can be substituted whole — no need to find x and y individually.
The answer is y = 4. Substitution is the whole method: replace x by its value in the second equation, giving 2 times 2, which is 4 rather than 2 plus 2.
Use the rank normal form PAQ = D with D the r-block identity. Taking B = QD⁻P, where D⁻ is the n×m block identity, gives ABA = A for every matrix A.
The answer is 7x^2 - 21x + 21. Expand the product with FOIL and the square with the (a - b)^2 rule, then add the two results together term by term.
The result is x^6 + 2x^5 + 3x^4 + 3x^3 + 3x^2 + 2x + 1. Squaring produces the palindromic pattern 1, 2, 3, 4, 3, 2, 1 and subtracting x^3 drops the middle 4 to 3.
The product is 40x⁶ − 68x⁴ − 58x³ − 24x² − 91x − 72. Distribute each of the three left terms over the trinomial, then collect the nine results by degree.
(4x² − 3)² expands to 16x⁴ − 24x² + 9. Use the pattern (a − b)² = a² − 2ab + b² with a = 4x² and b = 3 instead of multiplying the terms out one pair at a time.
(515 - 0.27x) * 98^2 = 4946060 - 2593.08x. Square 98 to get 9604 first, then distribute: 515 * 9604 = 4946060 and 0.27 * 9604 = 2593.08 for the x term.
The product is 45x^4 + 54x^3 - 40x^2 - 48x. Both left-hand terms multiply both right-hand terms, and since all four degrees differ nothing combines.
The product is 35x^4 + 7x^3 - 39x^2 + 34x - 10. Distribute each of the three left-hand terms across the trinomial, then collect like powers of x.
(98 - 0.18x) * 270^2 = 7144200 - 13122x. Squaring 270 gives 72900, then 98 * 72900 = 7144200 and 0.18 * 72900 = 13122, a whole-number x coefficient.
The numerator expands to a1x^2 - 2a1xy + (a1+a2)y^2 + b1x + (b2-b1)y. The nested form is Horner style, so unfolding it one bracket at a time is the safe route.
The cubic is -0.013122x^3 + 50.058x^2 - 57477.78x + 18572960. Multiply two binomials first, then distribute the third, tracking each decimal coefficient.
The answer is m^2 - 4m + 4. The middle term is twice the product of m and 2, and it is negative; the last term is positive because a square is never negative.
The standard-form polynomial is 20x^3 - 6x - 42. Distribute each product, watch the double negative in -6 times -3x^3, then collect the two cubic terms.
The standard form is x^2/3 - 4x/3 - 5/3. Square the bracket first, distribute the 1/3, then combine 4/3 with -3 by writing 3 as 9/3 to get -5/3.
The answer is x^2/3 - 4x/3 - 5/3. Multiply the brackets first to get x^2 - 4x - 5, then distribute the 1/3 across all three terms, not just the first.
The answer is pq - p - q + 2. FOIL gives pq - p - q + 1, and the extra 1 outside the bracket lifts the constant to 2 rather than cancelling it.
The answer is x^4 - 8x^(5/2) + 24x - 32/root x + 16/x^2. The fractional exponent makes each term drop by 3/2 rather than by 1, which is the easy slip.
The result is −8x³ − 48x² − 104x − 352, or −8(x³ + 6x² + 13x + 44). The quartic terms cancel because both binomials are raised to the same even power.
The product is x^2 + 7x + 10. FOIL gives x^2, 5x, 2x and 10, and the two middle terms add to 7x because 2 and 5 sum to 7 and multiply to 10.
The product is x^4 + 4x^3 - 12x^2 - 32x + 64. Both factors are perfect squares, so the product collapses to (x^2 + 2x - 8)^2 and needs only one squaring.
Answer: x = 0 and x = 1. Write 9 as 3^2 and move the fraction upstairs, then the substitution t = 3^(x-1) turns it into the quadratic 3t^2 - 4t + 1 = 0.
The solution is -2 < x <= 6. Substitute t = 2^x, combine both sides over a single denominator, read the sign chart, then convert the bounds on t back into x.
The solution is ((3-sqrt17)/2, 0) union ((3+sqrt17)/2, infinity). Match the bases, flip the inequality because 2/5 < 1, then sign-chart (x^2-3x-2)/x.
The solution is 2 <= x < 4. Factor the numerator by grouping into (2^x - 2)(5^x - 25), factor the quadratic denominator, then read a four-region sign chart.
The solution is 0 < x <= log7(2) or log2(7) <= x < 4. Group 14^x = 2^x*7^x into (7^x - 2)(2^x - 7), factor -x^2 + 4x, and read the sign chart.
The solution is x = 0 or 1 < x < log2(3). Substituting t = 2^x collapses the three fractions into 2(t-1)^2/((t-2)(t-3)), a perfect square over a quadratic.
The answer is (1/16)(x^3 + 8)^2. Substituting u = x^3 turns it into a quadratic that is a perfect square, and x^3 + 8 splits further as (x+2)(x^2-2x+4).
The factorization is (2x + 1)(5x − 4). Since 10 × (−4) = −40 and 5 + (−8) = −3, split −3x into 5x − 8x and group to pull out the shared binomial 2x + 1.
The factorization is (5x - 4)(25x^2 + 20x + 16). Write the two terms as (5x)^3 and 4^3, then apply the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2).
The answer is 6(2x + 3). The GCF of 12 and 18 is 6, not 2 or 3, and after dividing both terms by 6 the bracket 2x + 3 has no common factor left.
The GCF is 4, leaving 3x^2 - 44x + 120, whose discriminant 496 is not a perfect square - so the roots are (22 plus or minus 2 root 31)/3, not 6 and 20/3.
18abc^2 - 12a^2b^2c = 6abc(3c - 2ab). The greatest common factor pairs gcd(18, 12) = 6 with the lowest power of each shared variable, namely a, b and c.
Answer: (3x - 7)(7x^2 - 6). The rational root search misses it because the only rational root is 7/3; grouping the terms in pairs finds the factorisation instantly.
The answer is (x - 2)(2x + 1)(x + 1). Grouping fails here, so the Rational Root Theorem finds x = 2, and synthetic division leaves 2x^2 + 3x + 1.
The answer is x(2 - 9y). Only x is shared by both terms, because 2 and 9 have no common factor and the variable y appears in the second term alone.
The factorisation is 2(x - 6)(x + 1). Every coefficient is even, so pulling out 2 leaves the monic x^2 - 5x - 6, which factors with the pair -6 and 1.
The factorisation is (x - 2)(2x + 1). Multiply 2 by -2 to get -4, split -3x as -4x + x, then group the four terms and pull out the shared binomial.
The factorisation is (2x + 5)(x - 4). The AC product is -40, and the pair 5 and -8 sums to -3, which splits the middle term ready for grouping.
Answer: not over the integers, but yes over the reals: 2x^2 - 4x + 1 = (root2(x - 1) - 1)(root2(x - 1) + 1). The discriminant 8 is positive but not a perfect square.
The factorization is (2x - 1)(x - 3y). Grouping the first two terms as x(2x - 1) and the last two as -3y(2x - 1) exposes a shared bracket to pull out.
The answer is 2x(x + 2). The greatest common factor combines the numeric part 2 and the variable part x, so both come out together, leaving x + 2 behind.
The factorisation is 8x(4xy^2 - 3). The numerical GCF of 32 and 24 is 8, and only one power of x is shared, since the second term has no y at all.
The factorisation is (x + 1)(x + 2)(3x - 2). Testing x = -1 gives zero, and the cubic quotient 3x^2 + 4x - 4 then splits by the AC method with 6 and -2.
It factors as (3x^2 + 12x + 7)(x^2 + 4x + 9), giving real roots (-6 +/- sqrt 15)/3 and the complex pair -2 +/- i sqrt 5. Always expand a guessed factorisation back.
The answer is -B(2A - B)^2. After pulling out B the bracket is the negative of a perfect square, so a minus sign has to come out along with it.
The factorisation is 4(m - (1 + sqrt 6)/2)(m - (1 - sqrt 6)/2). The AC product -20 has no pair summing to -4, so the roots come from the quadratic formula.
The answer is 2mn(2m - 4n - 1). The last term is the whole GCF itself, so it leaves a -1 behind inside the bracket rather than disappearing.
The answer is (2x - 3)(2x + 3). Both terms are perfect squares, since 4x^2 is (2x)^2 and 9 is 3^2, so the roots of 4x^2 - 9 = 0 are x = 3/2 and -3/2.
The answer is y(2x + 3)^2. Pull out the common y first, and what remains is a perfect square trinomial because 12x equals 2 times 2x times 3.
5x⁴ − 80 factors as 5(x − 2)(x + 2)(x² + 4). Pull out the 5, apply the difference of squares twice, and stop once the sum of squares x² + 4 is left irreducible.
The factored form is 3(x − y)²(x + y). Collapse the second term to 3(y − x)³, use (y − x)³ = −(x − y)³ to align both terms, then take out 3(x − y)².
The factorization is 3(5a³ − 7b⁸)(5a³ + 7b⁸). Pull out the GCF 3 first, then 25a⁶ and 49b¹⁶ become perfect squares (5a³)² and (7b⁸)², so difference of squares applies.
7x² − 7x − 140 factors as 7(x − 5)(x + 4). Remove the common factor of 7 first, then split the remaining trinomial x² − x − 20 into two binomial factors.
81x² − 121y² factors as (9x − 11y)(9x + 11y). Rewrite the two terms as (9x)² and (11y)², then apply the identity a² − b² = (a − b)(a + b) directly to both.
The complete factorization is 9x(x - 1)(x + 1). The greatest common factor is 9x, not just 9, and the leftover x^2 - 1 is a difference of squares.
The factorisation is (3x + 1)^2. Both outer terms are squares, and the middle term 6x is exactly twice the product of 3x and 1, which confirms the pattern.
The expression collapses to (a²+ab+b²)(x²+xy+y²). Expanding A², AB and B² and collecting x², xy and y² terms gives the same factor a²+ab+b² three times.
The cyclic sum factors as −(a − b)(b − c)(c − a)[3(a² + b² + c²) + 5(ab + bc + ca)], a much sharper result than the fully expanded degree-5 form.
The factored form is (a − b)(a + x)^m (b + x)^(n−1). Take the lowest power of each base as the common factor; the bracket (a + x) − (b + x) collapses to a − b.
The answer is (a-b)(a+x)^m(b+x)^(n-1). Pull out the lowest power of each base, subtract inside the bracket, and the x terms cancel to leave a-b.
The answer is (a^2 - 2ab + 2b^2)(a^2 + 2ab + 2b^2). Adding and subtracting 4a^2b^2 creates a perfect square, turning a sum into a difference of squares.
It equals (a^5 - 1)/(a - 1) and is irreducible over the rationals - but not over the reals, where it splits into two quadratics with golden-ratio coefficients.
The factored form is (1/12)x^2y^2(6x^2 + 4xy + 3y^2). The fractional GCF is 1 over the LCM of 2, 3 and 4, which clears every denominator at once.
The answer is (a^2+ab-1)(ab+b^2-1). Expand first, then notice ab+b^2-1 divides it, giving the symmetric pair (a(a+b)-1)(b(a+b)-1) shown here.
Answer: (x^2 + 6x - 6)(x^2 - 6). Pair the brackets so both products share the constant -6, then substitute y = x^2 + x - 6 to get a simple quadratic in y.
The factorization is (y′ − 2x² − x)(y′ − 2x² + x). Grouping the first three terms into (y′ − 2x²)² turns the expression into a difference of two squares.
It factors as 2(4x^2 - 3x + 3)(x^2 + 3), so the roots are (3 +/- i sqrt 39)/8 and +/- i sqrt 3. Both quadratics have negative discriminants, so there is no real solution.
Answer: (x+2)(3x^2-5)(x^2+x+1). Find the root x = -2, divide it out synthetically, then split the quartic 3x^4+3x^3-2x^2-5x-5 into two quadratic factors.
The answer is (a^2 + b^2)(a^4 - a^2b^2 + b^4). Rewrite a^6 as (a^2)^3 so the sum of cubes formula applies, even though the exponent 6 is even.
The factorization is (a - 2x + 3)(a + 2x - 3). Grouping the three x-terms reveals a perfect square, turning the whole expression into a difference of squares.
The factorisation is (a + 2)^2. The constant 4 is 2 squared and the middle term 4a equals 2 times a times 2, so the square-of-a-sum pattern applies exactly.
A sum of squares does not factor over the reals, but over C it becomes (a + bi)(a - bi), because the cross terms cancel and i squared equals minus one.
Answer: (x-1)^2 (x^2-3x+1). Expanding gives the palindromic quartic x^4-5x^3+8x^2-5x+1, which splits as (x^2+ax+1)(x^2+bx+1) with a = -2 and b = -3.
The factorisation is (x - 1)(x - 2) and the roots are x = 1 and x = 2. A positive constant with a negative middle term forces both factors to be negative.
The roots are x = 1 and x = 3. Find two numbers multiplying to +3 and adding to −4, giving (x − 1)(x − 3), then apply the zero-product property to each factor.
The factorization is (x + y)(4x²y² + a² + 12). Pairing the six terms three ways exposes the binomial x + y hiding inside each pair, which is then collected out front.
The factorization is (f - 4g + 3h)(f + 4g - 3h). The last three terms are exactly -(4g - 3h)^2, so the expression is the difference of squares f^2 - (4g - 3h)^2.
The complete factorization is -x^2 y(x - 1)(x - 5). Take out the GCF x^2 y first, then pull the leading minus out of -x^2 + 6x - 5 before splitting it.
The complete factorisation is 3x^6(2x^3 - 1)(2x^3 + 1). Taking out 3x^6 leaves 4x^6 - 1, which is still a difference of squares and must be split again.
The answer is (3c/4 - 1)(3c/4 + 1). The fraction 9/16 is a perfect square because both 9 and 16 are, so its square root is the fraction 3/4.
The factorisation is (1/7 - c^5 n^3)(1/7 + c^5 n^3). Both terms are squares, since 1/49 is (1/7)^2 and c^10 n^6 is (c^5 n^3)^2 by halving the exponents.
27x^3+y^3=(3x+y)(9x^2-3xy+y^2), x^3-8y^3=(x-2y)(x^2+2xy+4y^2), x^3+512=(x+8)(x^2-8x+64). Learn to spot the two perfect cubes and write the factors.
The answer is (x - 1)(x - 2)(x + 3). Test the divisors of 6 to find the root x = 1, divide out x - 1, then factor the leftover quadratic x^2 + x - 6.
The factorisation is (x - 1)(x + 2)^2. Testing x = 1 gives zero, synthetic division leaves x^2 + 4x + 4, and that quotient is a perfect square.
The answer is (x + 3)(x - 2)(x + 2). Grouping in pairs exposes the common factor x + 3, and the remaining x^2 - 4 is a difference of squares.
The answer is (x + 1)(x^2 + 2x + 5). Only x = -1 works among the rational candidates, and the quadratic factor has discriminant -16, so it stops there.
Over the rationals it is (x - 1)(x^2 + 5x + 5). The quadratic factor has discriminant 5, so only over the reals does it split further using (-5 +- sqrt 5)/2.
The answer is (x + 1)(x - 2)(x + 2). Grouping in pairs exposes the shared factor x + 1, and the leftover x^2 - 4 is a difference of squares.
The complete factorisation is (x + 1)^2 (x - 1). Grouping gives (x + 1)(x^2 - 1), and the second factor is itself a difference of squares that must be split.
It factors as (x + 1)(x^2 + x + 1)(x^2 - x + 1). The sum is the geometric series (x^6 - 1)/(x - 1), so factoring x^6 - 1 twice and cancelling x - 1 ends it.
The answer is (x + y)(x^4 - x^3y + x^2y^2 - xy^3 + y^4). A sum of powers factors only when the exponent is odd, and the long factor's signs alternate.
The factorization is (x² + xy + y²)². The palindromic coefficients 1, 2, 3, 2, 1 are the signature of squaring a three-term expression rather than a binomial.
The answer is (x^2 + 2x + 3)(x^2 - x + 1). The quartic has no rational roots, so match coefficients against (x^2+ax+b)(x^2+cx+d) with bd = 3 instead.
It factors as (x^2 - x + 1 - a)(x^2 + x + 1 + a). Completing (x^2 + 1)^2 rewrites the expression as (x^2 + 1)^2 - (x + a)^2, a plain difference of squares.
It equals 2(x - 2)(x + 2)(x^2 + 10). Adding the two fourth powers cancels every odd-degree term, leaving 2x^4 + 12x^2 - 80, which is a quadratic in x^2.
The factorization is 2x(x+4)(x^2+4x+14). Centering the substitution at y = x + 2 kills the odd powers, leaving a biquadratic in y that factors as a quadratic in y^2.
The simplified form is (x + 5)/(x + 6)^(6/5). With negative exponents the smaller one is -6/5, so that is the power you factor out of both terms.
The expression factors to (x + 5)/(x + 6)^(6/5). Take out the smaller power (x + 6)^(−6/5) as the common factor, leaving (x + 6) − 1 = x + 5 inside.
The simplified form is -(x + 6)(x + 7)^(1/3). Factor out the lower power (x + 7)^(1/3), then simplify the leftover bracket 1 - (x + 7) into -x - 6.
The answer is (x - 64)(x - 36), and not (x - 48)^2. Evaluate 48^2 = 2304 first, then find the negative pair -64 and -36 that adds to exactly -100.
The factorization is (x - a - 1)(x - a + 2). Substituting u = x - a collapses the expression to u^2 + u - 2, which splits at once as (u - 1)(u + 2).
The answer is (x - y - 3)(x - y + 3). The first three terms collapse to (x - y)^2, and what remains is then a difference of squares against 3^2.
The answer is x(x - 4). Both terms contain x, so x is the greatest common factor; pulling it out leaves x - 4, and the roots are then x = 0 and x = 4.
The answer is (x - 3)(x - 4). A positive constant with a negative middle term means both numbers are negative: -3 and -4 multiply to 12 and add to -7.
The discriminant is -3, so there is no real factorisation. Over C it splits as (x - (1 + i root 3)/2)(x - (1 - i root 3)/2), a conjugate pair of roots.
x² + 25 is prime over the real numbers. A sum of squares has no real roots, so unlike x² − 25 it cannot be split into a product of two real linear factors.
The factorization is (x - 4y)(x + y + 2). Factor the degree-two part as (x - 4y)(x + y) first; the leftover 2x - 8y = 2(x - 4y) then shares that same bracket.
The answer is (x + 4)(x - 3). A negative constant forces opposite signs, and 4 with -3 is the only pair that multiplies to -12 and adds to +1.
The answer is (x + 3)(x - 2). The negative constant forces opposite signs, and 3 with -2 is the only pair multiplying to -6 while adding to +1.
The answer is (x + a)(x - a + 1). The two numbers a and 1 - a multiply to a - a^2 and add to 1, which is exactly what the factoring pattern needs.
The discriminant is -3, so there is no real factorisation. Over C it is (x + 1/2 - i root 3/2)(x + 1/2 + i root 3/2), whose roots are cube roots of unity.
Answer: (x^2 - xy + y^2)^2. Expanding gives the symmetric quartic x^4-2x^3y+3x^2y^2-2xy^3+y^4, which factors as a square once you try (x^2+axy+y^2)(x^2+bxy+y^2).
The answer is (x - y)(x - 1)(x + 1). Since y - x = -(x - y), the two brackets match up, and the leftover x^2 - 1 is a difference of squares.
The factorisation is y(x - 2)^2. Pulling out the common y leaves x^2 - 4x + 4, a perfect square, so the answer needs both a GCF step and an identity step.
The answer is (x - 1)(x^104 + x^103 + ... + x + 1). A difference of nth powers always has x - 1 as a factor, whatever n is, and the cofactor is all plus signs.
The answer is (x + 1)(x^104 - x^103 + ... - x + 1). Because 105 is odd, x = -1 is a root, so x + 1 divides the expression and the cofactor alternates in sign.
The factorisation is (x^2 - 1/2)^2. Treating x^2 as the variable turns the quartic into a perfect square, since 1/4 is (1/2)^2 and -2 times x^2 times 1/2 is -x^2.
The answer is (x - y)(x^n-1 + x^n-2 y + ... + y^n-1). The cofactor's signs are all positive, and the middle terms telescope away when the product is expanded.
The answer is (x - y)(x + y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)(x^4 - x^3y + ...). Split as a difference of squares first, then use the fifth-power formulas.
Answer: (x-1)^2 (y-1)^2. Expanding everything gives a nine-term expression that is exactly the square of xy - x - y + 1, which itself factors as (x-1)(y-1).
The answer is ab = -1/4, from a = -1/2 and b = 1/2. The solution set lying between the roots forces a < 0, and the constant term 1 pins down the scale factor.
The answer is a ≤ 1. Substituting x = 3 gives 2a, and "3 not in A" means 2a > 2 must be false, so 2a ≤ 2 — no sign chart is needed anywhere.
f(2) = 18. Writing (x + 1)f(x) = (x^3 + 2)Q(x) + x + 2 with Q linear, the root x = -1 and f(0) = 4 pin down Q(x) = 2x + 1.
k = 3. The function is point-symmetric about (5, 7) and the bounds 1 and 13 are symmetric about 7, so the interval k − 9 ≤ x ≤ 3k + 7 must be centred at x = 5.
k = 5. The inequality is symmetric about x = 4, so the interval midpoint (k - 8 + 2k + 1)/2 must equal 4, giving 3k - 7 = 8 and the solution set -3 < x < 11.
The answer is 2022 ≤ λ ≤ 2026. Only the square root constrains λ: its radicand needs (λ − 2024)² ≤ 4, which is |λ − 2024| ≤ 2 centred on 2024.
m = 13. Substituting x = 2 turns 2mx into 4m, so the equation becomes 36 + 4m = 88; subtracting 36 then leaves 4m = 52, and dividing by 4 gives m = 13.
The answer is 5/4. Divide to isolate the conjugate of z-1, rationalise the denominator, conjugate both sides to recover z, then use |z|^2 = a^2 + b^2.
N = 42u²v − 25uv² + 3v³. Add the subtracted trinomial to both sides, then combine like terms carefully — the uv² terms are −10 and −15, giving −25.
P = 12m²n + 3mn² + 27mn. Because P is the one being subtracted, it equals the first polynomial minus the second, so every sign in the second one flips.
Answer: |Z| = 2. Write Z = r e^(i theta); the common factor e^(-i theta) has modulus 1, leaving root(r^2 + 16/r^2) = 2 root 2 and the perfect square (r^2 - 4)^2 = 0.
The other interval is (-infinity, -3). Because x = 2 must be a root, b = 6, and x^2 + x - 6 = (x + 3)(x - 2) is positive outside its two roots.
Answer: (-infinity, -3), with b = 6. Since (2, +infinity) is a solution interval, x = 2 is a root, giving b = 6 and the factorisation (x+3)(x-2) > 0.
The reciprocal of 1/(5y²) is 5y², valid for y ≠ 0. Flipping a fraction swaps numerator and denominator; the check is that the product of a value and its reciprocal is 1.
Answer: y = 2254 and x = 2231.46. Clear both denominators to get x = 0.99y and x = 0.98(y+23), equate them, and a coefficient of only 0.01 is left over.
Solving 2018 - 5(n - 1) < 0 gives n > 404.6, so the first negative term is u_405 = -2. The previous term u_404 = 3 is still positive, confirming the cutoff.
Answer: divide 8x^3 by 2x, giving 4x^2. Polynomial long division always starts by dividing the leading term of the dividend by the leading term of the divisor.
Solving 3 + 7(n - 1) > 2018 gives n > 288.86, so n = 289 and u_289 = 2019. The previous term u_288 = 2012 is still below 2018, pinning the answer.
The solution is x = 17, y = 20, z = −13, w = −13/3. Clearing the fractions early and chaining the substitutions x = 30 + z and w = z/3 reduces everything to one equation in z.
The graph is an upward parabola with vertex (0, 1) and axis x = 0 - the parent y = x^2 shifted up one unit. Its range is y >= 1, so it never meets the x-axis.
The vertex is (−5, −4) and the V opens downward with slopes ±1, maximum −4, y-intercept −9, and no x-intercepts since every output is at most −4.
The solution is the filled diamond with vertices (4,0), (0,3), (-4,0) and (0,-3). Intercepts plus double reflection symmetry give the whole region quickly.
The solution is the closed upper half-plane. Because y >= 0 has no x in it, the boundary is the x-axis itself and the shading covers everything at or above it.
Answer: for k not equal to 1 only (0,0,0); for k = 1 the whole line (0,t,t). Eliminating between equations 1 and 3 forces x = 0, then (k-1)z = 0 splits the cases.
Answer: (x, y, z) = (0, t, t) for any real t. The determinant is zero, so besides the trivial solution the system has a whole line of them through the origin.
The APs 2, 5, 8, ... and -1, 6, 13, ... (1000 terms each) share terms with x = 2 mod 3 and x = 6 mod 7, i.e. x = 20 mod 21 up to 2999, giving 142 values.
The graph of g is the graph of f translated 9 units straight down, moving the vertex from (0, 0) to (0, -9). The V shape and both slopes stay unchanged.
At the origin both z = 0 and z = 2 satisfy the equation, so no single-valued f exists. The surface is x^2 + y^2 = sqrt(8z) - z^2 for 0 <= z <= 2.
The solution is h > 1.3397214. No h ≤ 0 works; for h > 0 the inequality is equivalent to h⁴ − 1.8h − 0.81 > 0, whose single positive root is found by Newton's method.
The answer is x ≤ 1. Multiplying through by the LCD 6 removes both fractions, and dividing by −11 at the very end is the step that flips the inequality sign around.
The solution is x < 2 with x ≠ −1. Factoring gives (x + 1)²(x² − x + 1)(2 − x) > 0, and since the square and the quadratic are never negative, only 2 − x sets the sign.
A 1001-term AP from 1 to 2018 has d = 2017/1000. The 501st term is 1 + 500d = 1 + 2017/2 = 2019/2 = 1009.5, exactly the midpoint of 1 and 2018.
Placing 3 numbers between 2 and 22 makes a 5-term AP, so 22 = 2 + 4d and d = 5. The inserted means are 7, 12 and 17, giving 2, 7, 12, 17, 22.
Answer: (0.4433, 0.7268) and (-0.6025, 4.9100). Setting the two expressions equal and clearing the denominator gives the quadratic 9.36x^2 + 1.49x - 2.5 = 0.
Answer: about (0.4569, 0.6724) and (-0.9500, 6.2998). Clear the denominator to get 288x^2 + 142x - 125 = 0, solve, then check neither root hits the pole.
Answer: t inverse(x) = x^2/9 - 6 for x >= 0. Swap x and y, divide by 3 before squaring, and the restriction comes from the original range being non-negative.
No, it is wrong: a comma in set-builder notation means AND, so the set is empty. To describe [2, 5] ∪ {10} you must write the word or, as in {x | 2 ≤ x ≤ 5 or x = 10}.
There are four solutions, including the exact pair (3/5, 4/5, -5) and (-1, 0, 1); the other two involve sqrt 11. Every candidate must satisfy x^2 + y^2 = 1.
x = 6. When y = 2x^2 - 21x + 64 and y = 3x + a meet exactly once the quadratic has a double root, so x = 24/4 = 6 and a = -8. Full discriminant walkthrough.
The line is y = −4x + 3. Use point-slope form, watch the double negative x − (−1) become x + 1, then convert to slope-intercept form and check the point.
The equation is y = (2/3)x + 9. Start from point-slope form y − 5 = (2/3)(x + 6); the fraction clears cleanly because the x-coordinate −6 is divisible by 3.
x ≈ 128.0002, and the intended exact answer is 128. Both decimals are rounded versions of 3/110 and 384/110, whose quotient is exactly 128.
The solution is X = −10.26, or −513/50. Combine 2000/180 = 100/9 with 5/3 into a single coefficient 500/27, subtract 2210, then multiply by the reciprocal.
The solution is x = -3. Every term already has denominator 3, so each side collapses to a single fraction and the equation reduces to x - 1 = 2x + 2.
Answer: all real numbers. Both sides expand to 3 - 8x, so the equation reduces to 3 - 8x = 3 - 8x, an identity rather than a condition that pins down x.
The solution is u = −1127/300 ≈ −3.7567. Reduce 4520/1200 to 113/30, write 0.01 as 1/100, then subtract over the common denominator 300.
Answer: x = -209/7, about -29.857. Combine each side into a mixed number, subtract to isolate the coefficient 7/45, and let the common denominator 45 cancel.
The equation has no solution. Clearing the nested brackets gives 6x - 23 = 6x - 2; the x terms cancel and leave the false statement -23 = -2.
The solution is x = 7250 exactly. Distribute 0.03 to get 0.024x − 150, flip the sign of the whole bracket, collect to 0.976x + 150 = 7226, then divide.
The solution is x = 442625/61 ≈ 7256.15. The same expansion gives 0.976x + 150 = 7232, but 7082/0.976 does not terminate, so the exact answer is a fraction.
Answer: x = 1,646,872.17, exactly 2653438800000/1611199. Factor x out of the left side, combine the bracket into one fraction, then multiply by its reciprocal.
The solution is x = 29300000/171 ≈ 171345.03. Multiply 0.0005 by 52 to get 0.026, collect 1x + 0.026x = 1.026x, then divide 175800 by 1.026.
Answer: x > 80/3, about 26.67. Subtract x to get 0.3x - 5 > 3, add 5, then divide by the small positive coefficient 0.3 - not by 2 as for 3x - 5 > x + 3.
The answer is n >= -23/6. Distribute the -2, combine constants to 29, add 8n to both sides, then divide by the positive 6 so the sign never flips.
The answer is n >= -5. Distributing -3 gives 31 - 3n, and after adding 8n the constants land on a whole number, so the bound is an integer, not a fraction.
Answer: x > -543/13, about -41.77. Simplify each side to 3x - 299 and 8.2x - 81.8, then divide by the positive 5.2 so the inequality sign is unchanged.
Answer: x > 10. Multiply by the LCD 20 to reach 5(x+2) - 4x > 20, expand to x + 10 > 20, and subtract; the x terms nearly cancel, leaving a single step.
Answer: x > 6. The trap is 4 - (x-2)/2 = 5 - x/2, not 3 - x/2. Collect the x terms over the denominator 6 to reach 5x/6 > 5 and finish in one division.
Answer: x < 11. Multiply both sides of (2x-1)/3 > (3x-5)/4 by the positive LCD 12, distribute to reach 8x-4 > 9x-15, then isolate x; no sign flip is needed here.
The answer is the empty set. Subtracting x from both sides erases the variable and leaves -1/3 > 4, a false statement, so no value of x can ever work.
The solution is x <= -1/2. Combine each side over a common denominator, multiply through by 12 to clear fractions, then isolate x from 14x - 2 <= 12x - 3.
Answer: x > -1. Multiply by the positive LCD 10 to get 5(x-1) < 2(7x+2), expand to 5x-5 < 14x+4, and move the x terms so the coefficient stays positive.
log y = b log x + log a. Because the unknown is the power, both axes need logs: gradient b and intercept log a come straight off a log-log graph.
log y = -(log p)x + log q. Plotting log y against x (not log x) gives a straight line of gradient -log p and intercept log q.
1/y = (b/a)(1/x) - 1/a. No logarithm is needed here: the variables are tangled in a product, so reciprocals - not logs - straighten the graph.
The value is exactly 1. Because 18 = 3^2 * 2, the product rule gives log_3(18) = 2 + log_3(2), so the numerator and denominator are literally the same number.
The equation has the single root x = -3, so the sum of its roots is -3. Change of base turns the quotient of logs into one logarithm with base x + 5.
The solution is x ≤ −5/2, x = 0, or 1/6 ≤ x < 1/2. Split on the sign of x, divide by x (flipping the sign when x < 0), and use that log base 1/3 is decreasing.
The quotient is 3x + 2 with remainder 3, so the result is 3x + 2 + 3/(x - 2). The nonzero remainder equals f(2) = 3, exactly as the Remainder Theorem predicts.
The quotient is 2x - 4 with remainder 2, i.e. 2x - 4 + 2/(3x + 2). Because the divisor leads with 3x rather than x, each quotient term needs a division by 3.
The quotient is 4x^2 + 7x + 15 with remainder 78, i.e. 4x^2 + 7x + 15 + 78/(2x - 5). The divisor is non-monic, so every quotient term comes from dividing by 2x.
The quotient is x^2 + 4x - 5 with remainder 0, so x + 3 is a factor and the cubic factors completely as (x + 3)(x + 5)(x - 1) with roots -3, -5 and 1.
Proof that if a square matrix commutes with all matrices of the same size, it must equal λI. Uses matrix units E_ij to force every off-diagonal entry to zero.
The answer is x = (36 - sqrt 71)/12. Cross-multiplying gives 6x = a^2, and squaring the surd produces 72 - 2 sqrt 71 before dividing by 4 and then 6.
The minimum is (4√2 − 3)/6 ≈ 0.4428, reached at a = (5 − 3√2)/2 and b = 3√2 − 4. Substituting b = 1 − 2a and setting the derivative to zero gives (1 + 2a)² = 8(1 − a)².
The minimum value is 3/16, reached when a = b = c. A power-mean step plus a two-term lemma pins the sum of squares below by 3/4, which is exactly enough.
Answer: a <= 7. The sum of four absolute values has minimum 7, attained on the whole middle interval [-1, 1] between the two inner points, not on [-3, 1].
Mixing 10 L of the 20% solution with 10 L of the 50% solution gives 20 L at 35% alcohol. Set up the volume and pure-alcohol equations and solve the system.
Answer: 25. Modulus rules collapse the expression to |Z2|^2, so only Z2 = (7i^88 + 24i^33)/(5i) matters; it simplifies to 24/5 - 7i/5 with |Z2| = 5.
The value is 27. Cancelling Z2 reduces the expression to |Z1|^3, and the two given equations force conjugate Z1 = -3i, so Z1 = 3i and |Z1| = 3.
The product is 21x^4 + 28x^3y + 62x^2y^2 + 36xy^3 + 45y^4. Sorting the six partial products by total degree in x and y shows only the x^2y^2 pair combines.
(4x + 1)(3x + 2) equals 12x² + 11x + 2. FOIL multiplies the First, Outer, Inner and Last pairs, then the two middle terms 8x and 3x combine into 11x.
The product is 10x^4 + 24x^3 - 63x^2 + 27x. Distribute 5x^2 and then -3x across the trinomial, then combine the two like x^3 terms and the two like x^2 terms.
(6x² − 7)(5x² − 6) equals 30x⁴ − 71x² + 42. Treat x² as a single unit, multiply the four pairs of terms, then combine −36x² and −35x² into the middle term −71x².
The product is 56y^6 + 21y^5 - 21y^4 - 56x^2y^3 - 35y^3 + 21y^2 - 56xy^2. Multiply coefficients and add exponents on y for each of the seven terms.
The product equals 2 times the cube root of 3. Combine the two radicals into the cube root of 24, then pull out the perfect cube factor 8 that divides 24.
The product is -18x^4 + 8x^3 + 59x^2 - 28x + 14. The first factor has no x term, so only the x^2 column receives contributions from both halves of the distribution.
The product equals 2x² − 4x + 1. Group the x-coefficients into the conjugate pair 2 ± √2, use (2−√2)(2+√2) = 2 for the x² term, and the surds cancel in the x term.
The product is x³ + 1. Distributing gives x³ − x² + x + x² − x + 1, where four terms cancel — the shortcut is the identity a³ + b³ = (a + b)(a² − ab + b²) with a = x, b = 1.
The product is 2x^2 + 9xy - 5y^2. Distributing gives four terms, and the two middle terms -xy and +10xy combine into the single like term 9xy.
The product is 2x^4 + 6x^2y^2 + 6x^2z^2 + 4y^4 + 6y^2z^2. Only the x^2y^2 terms combine; the other four monomials involve different variable patterns and stay separate.
Answer: no real solution. A sum of nonnegative squares is zero only if both vanish, and 1.6 does not equal 1.5; expanding confirms a discriminant of -0.08.
The answer is 15 pens. Set the price as 240/x, write the condition 240/x − 240/(x+1) = 1, clear the denominators to x² + x − 240 = 0, and discard the negative root.
Rows of 1, 2, 3, ... trees give the triangular sum n(n + 1)/2 = 3003, so n^2 + n - 6006 = 0 and n = 77. Checking, 77(78)/2 = 3003 exactly.
The answer splits into seven cases. The sign of a decides whether the solution is a bounded interval, two rays, all of R or empty, and b decides the boundary.
After moving 2x across, the inequality factors as (x−2)(ax−1) ≥ 0. The answer splits at a = 1/2: outside the roots 2 and 1/a, with all of ℝ when the roots coincide.
The discriminant is the perfect square (3a + 4)², so the roots are always −3 and 4/a. The solution set splits by the sign of a, with a = 0 and a = −4/3 as separate cases.
The decomposition is 1/n - 2/(2n + 1). Factoring the denominator as n(2n + 1) gives two distinct linear factors, and comparing coefficients yields A = 1, B = -2.
The result is 2/(x - 1) + 1/(x - 1)^2. A repeated linear factor needs one term per power, and matching coefficients gives A = 2, B = 1.
The decomposition is 1/(x - 1) + 2/(x - 1)^2. A repeated linear factor needs one term per power, and equating coefficients gives A = 1 and B = 2.
The factorization is (3x − 2y)². Both end terms are perfect squares, 9x² = (3x)² and 4y² = (2y)², and the middle term −12xy is exactly −2(3x)(2y).
x^2 - 8xy + 16y^2 = (x - 4y)^2. Check that 16y^2 is (4y)^2 and that the middle term equals -2 times x times 4y, then read off the square of a difference.
The perimeter is 40 feet. Adding the three expressions gives 3x^2 + 3x - 20, which at x = 4 evaluates to 40 - combining like terms first is faster.
The difference simplifies to 12a^2b^2 - 5ab^5, a binomial of degree 6. See why subtracting cancels the a^3b terms and why degree comes from adding exponents.
The missing quotient entry is A = −3, so the quotient is x − 3 with remainder 0. Learn how each cell of a polynomial division table is generated, step by step.
The quotient is x − 2 and the remainder is 8x − 5. A quadratic divisor rules synthetic division out, so long division is the method that applies here.
The quotient is x and the remainder is −6x + 5, so the cubic equals x(x² + 2) − 6x + 5. Both the dividend and divisor have gaps that need placeholder zeros.
Expand (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 and 3xy(x + y) = 3x^2y + 3xy^2; subtracting cancels both mixed terms and leaves exactly x^3 + y^3.
The roots are x ≈ 142.379 and x ≈ 17.621, exactly (1360 ± 105√102)/17. The discriminant is 426147.84 − 167051.52 = 259096.32, whose square root is 509.01505.
The roots are x = (11 - sqrt29)/2 = 2.807 and x = (3 + sqrt13)/2 = 3.303. Split at x = 3, solve a quadratic on each branch, and keep the roots that fit.
Answer: 2 − 2√3 ≤ x ≤ 2 + 2√3. The roots give f(x) = a(x + 1)(x − 5), the minimum −18 at the vertex x = 2 forces a = 2, and then you solve 2x² − 8x − 10 ≤ 6.
The real solutions are x = −12 and x = −13. Substitute u = x + 4 to turn a quadratic-in-form equation into u² + 17u + 72 = 0, factor, then convert back.
The answer is -1/2 <= x <= 2/3. The hidden leading coefficient is -6, so the product is positive between the roots rather than outside them.
The answer is 3/2 < x < 2. The hidden leading coefficient is -2, so the product is positive strictly between the roots rather than outside them.
Answer: x < -1/2 or x > 2. Factor the non-monic quadratic as (2x + 1)(x - 2), find the critical points -1/2 and 2, and keep the outside intervals.
The answer is x <= -1/3 or x >= 1. The AC method splits -2x into -3x + x, factoring it to (3x + 1)(x - 1), and the parabola opens upward from there.
The only answer is x = 1/2. The trinomial is the perfect square (2x - 1)^2, which is never negative, so equality is the only way to satisfy the inequality.
Answer: no solution. The quadratic is the perfect square (2x - 1)^2, and a real square is never negative, so the strict inequality has an empty solution set.
The answer uses the roots (-8 plus or minus 8 root 10)/3, about 5.766 and -11.099. The trinomial does not factor over the integers, so x = 6 is not a root.
The answer is m < -6 or m > 2. Squaring -m removes the sign, and distributing -4 flips both inner signs, giving the factorable m^2 + 4m - 12.
There is no solution. Expanding and rearranging gives 3x^2 - 6x + 7 < 0, but the discriminant is -48, so this upward parabola stays strictly above the axis.
Answer: [-1, 0] union [2/3, 4]. Factor each quadratic, solve the two inequalities separately, then intersect the solution sets on one number line to finish.
Answer: [-1, 0] union [5/3, 4]. Multiply the first inequality by -1 to flip its sign, factor 3x^2-5x as x(3x-5), then intersect the two solution sets.
Answer: x < -2*sqrt(11363)/11 or x > 2*sqrt(11363)/11, about +/-19.3813. Evaluate the right side to 19285, divide by 55, and keep the bound as a fraction.
It factors as (x-1)(ax+1), so the answer needs five cases - not three. The a < 0 branch splits further at a = -1, where -1/a equals 1 and the roots coincide.
The solution set is (−∞, 1) ∪ (3, ∞). The roots 1 and 3 split the line into three intervals, and the product of two linear factors is positive only outside the roots.
The answer is x < -1 or x > 3. The parabola opens upward, so the product is positive outside the two roots and negative in the strip between them.
The solution is x < -1 or x > 3. Subtract 3 to reach (x - 3)(x + 1) > 0, then note that an upward-opening parabola is positive outside its two roots.
The answer is x < 1 - root 2 or x > 1 + root 2. The trinomial does not factor over the integers, so the quadratic formula supplies the two boundary points.
The answer is the open interval (-1, 3). The trinomial factors as (x - 3)(x + 1), and an upward parabola is negative only between its two roots.
The solution is x < -1 or x > 5. Factoring gives (x - 5)(x + 1) > 0, and testing one point in each of the three intervals shows where the product is positive.
The answer is -1 < x < 5. Factor to (x - 5)(x + 1), then use the upward-opening parabola: a positive quadratic is negative only strictly between its two roots.
The answer is all real x. The discriminant is -3, so there are no real roots, and completing the square gives (x + 1/2)^2 + 3/4, a positive minimum of 3/4.
The greatest possible value of n is 14. Factoring f(x) = a(x - 1)(x + 7) gives b = 6a, so a + b = 7a, and the smallest allowed integer a is 2, giving 14.
The answer is 7/4 < k < 13/6. Sign conditions f(0) > 0, f(1) < 0, f(2) < 0, f(3) > 0 pin the parameter down, and the product of the roots gives a check.
All four roots are real: m ≈ −0.86460, 3.37180, 11.10473, 13.31466. A sign scan finds every bracket, and Vieta's sum and product confirm none were missed.
Answer: no real roots. A sum-of-squares split, 3(x^2+3x+3/2)^2 + 2x^2 + 5x + 21/4, shows the quartic never drops below 17/8, so all four roots are complex.
The roots are u = −1, u = 1 and the double root u = 3. Matching coefficients gives (u² − 1)(u² − 6u + 9), which factors further as (u − 1)(u + 1)(u − 3)².
Answer: x = 0.0115484893 and x = -0.0115485279. The quartic term dominates, so the roots are nearly the fourth roots of 1210.23/6.804e10 and nearly symmetric.
Answer: x = 947.0356 and x = -1332.1428. A tiny leading coefficient is not a negligible term at large x: at x = 1729 the quartic piece is already about 6079.
The real roots are x ≈ 8.4610649 and x ≈ −0.3736462, plus a complex pair. This quartic has no rational root and no integer quadratic split, so bracket and use Newton.
The roots are x = 1, x = 7 and x = 4 ± √21. Both quadratic factors share x² − 8x, so setting t = x² − 8x turns the quartic into t² + 2t − 35 = (t + 7)(t − 5).
Answer: x = 205/18, about 11.3889. Divide by 3 before squaring, so root(2x - 1) = 14/3; then 2x - 1 = 196/9, giving 2x = 205/9 and finally x = 205/18.
The only solution is x = 3. Squaring produces x = 0 and x = 3, but x = 0 makes the right side negative, so it is extraneous and must be discarded.
Answer: x = 7/2 or x = -3/2. The radicand factors as 4(x - 1)^2, so the root is 2|x - 1|; the absolute value produces exactly two solutions.
Answer: x = 2. Squaring cancels the x^2 terms and leaves -3 = -2x + 1, so x = 2; it satisfies the domain condition x >= 1, so it is not extraneous.
Answer: no real solution. Squaring gives 3x^2 - 14x + 24 = 0, whose discriminant is 196 - 288 = -92, so no real x satisfies the squared equation either.
Answer: no solution. Squaring gives x^2 - 6x + 25 = x^2 - 6x + 9, so all the x-terms cancel and the equation reduces to the false statement 25 = 9.
Answer: x = 0 or x = 6. Squaring is safe because the right side 5 is positive; the constant 25 then cancels from both sides, leaving x(x - 6) = 0.
There are two solutions, x = 0.9 and x ≈ 0.8974432593. Substituting u = 8.9 − x turns the equation into 71(u−8) = (8.9−u)√((u−8)(u+8)), exposing the boundary root u = 8.
A column-space basis gives A = UC = sum of r outer products; subadditivity of rank shows fewer than r is impossible. Two halves, two different tools.
The solution is (1/4, 4]. Split the chain into two rational inequalities, move everything to one side of each, and intersect the sign-chart results while excluding x = -1.
x = 5/3. The common denominator (x - 2)(x + 2) is exactly x^2 - 4, which reduces the equation to 3x - 2 = 3, and 5/3 avoids the excluded values 2 and -2.
The solution is X = 21/47 ≈ 0.4468. Combine 450X − 6X into 444X, cross-multiply to 3150 + 54X = 7104X, then reduce 3150/7050 by its gcd 150.
Answer: x = (-225 +/- 25*sqrt(161))/2, about 46.107 and -271.107. The numerator collapses to the constant 2500, leaving x^2 + 225x - 12500 = 0 to solve.
The numerator collapses to -2500, giving x^2+225x+37500=0 with discriminant -99375, so no real solution exists: x = (-225 +/- 25i*sqrt(159))/2 over C.
There is no real solution. Combining the fractions collapses the numerator to the constant −2500, so the equation becomes x² + 225x + 37500 = 0 with discriminant −99375 < 0.
The answer is [2/3, 1). Move the 1 across and combine over one denominator to get (3x - 2)/(1 - x) >= 0 - and never cross-multiply by 1 - x.
The answer is the half-open interval [1/2, 1). The numerator's zero is included because 0 satisfies >=, but x = 1 is excluded since it kills the denominator.
Answer: [-1, 1/2]. The denominator has discriminant -7 so it is positive for every x, meaning the sign of the whole fraction is decided by the numerator alone.
The answer is the open interval (-1/2, 4). A quotient is negative exactly when numerator and denominator have opposite signs, which happens only between the roots.
The answer is -1/2 < x <= 2/3. Subtracting 1 gives (2 - 3x)/(2x + 1) >= 0, and the left end stays open because the denominator vanishes exactly there.
The answer is x < (-3-root 37)/2 or 1 < x < (-3+root 37)/2. Never cross-multiply: the sign of x - 1 is unknown, so combine into one fraction and use a sign chart.
Answer: 1 < x <= 2, the interval (1, 2]. Move everything to one side, combine into (6 - 3x)/(x - 1) >= 0, and use a sign chart - never multiply by x - 1 of unknown sign.
The solution is the interval (2, 5). Subtracting 1 and combining into (5 - x)/(x - 2) > 0 avoids multiplying by x - 2, whose sign is unknown.
Answer: x in ((-1 - root409)/6, -2) union ((-1 + root409)/6, infinity). Combine the polynomial parts into (3x - 5)/6, then sign-chart (3x^2 + x - 34)/(x + 2).
The answer is (-3, 1]. Both parts share the factor x - 3, which cancels but still removes x = 3 from the domain - a hole, not a sign change.
The answer is (17√11 − 34)/7. Multiplying by the conjugate 2 − √11 turns the denominator into 4 − 11 = −7, so the minus sign then moves up into the numerator.
The answer is (2 root 7 - 2 root 2) / 5. Multiply top and bottom by the conjugate root 7 - root 2, so the difference of squares clears both radicals.
The cube root of 7/y simplifies to the cube root of 7y² divided by y. Multiply inside the radical by y²/y² so the denominator becomes the perfect cube y³.
The cubic has three real roots: -2.7875545, 3.2205249 and 4.5670296. No rational root exists, since the only candidates are the divisors of 41, namely 1 and 41.
The answer is x ≈ 4.9807×10⁻⁵, not exactly 5×10⁻⁵. The right side is just 1.17% of 8×10¹², which shifts x by about 0.39% — small, but very far from zero.
The root is x = 4. Two reciprocals are equal only when their denominators are equal, so 4x − 5 = 11, giving 4x = 16 and x = 4, which is inside the domain.
The result is f(x) = −|x + 2| − 1. Reflecting negates the entire function, turning the −1 into +1, and the later shift down 2 brings the constant back to −1.
The result is f(x) = −|x + 2| + 1 with vertex (−2, 1). Reflection sends the −3 to +3, and the shift down 2 leaves +1 — the constant does not simply stay negative.
The result is f(x) = −|x| − 6 with vertex (0, −6) and arms of slope ∓1. Every output is negative, so the graph lies entirely below the x-axis with maximum −6.
The remainder is 26x - 47. If f(x) leaves remainder 2x + 1 on division by 2x^2 - 3x + 4, cube that remainder and reduce it using the rule x^2 = (3x - 4)/2.
Use the Remainder Theorem to divide x³ − 2 by x − 1 in one step, without long division. Includes why the remainder is a constant and not a polynomial.
Euler's formula splits each exponential into cosine and sine; the cosine parts add and the sine parts subtract, giving a clean real-plus-imaginary form.
The slope-intercept form is y = x/4 - 7/4. Distributing the 1/4 and combining 1/4 with -2 over the denominator 4 gives a slope of 1/4 and intercept -7/4.
The roots are x ≈ −29.572 and x ≈ 51.859, more than 81 apart. A tiny leading coefficient makes the parabola almost flat, pushing the roots far from the vertex at x ≈ 11.14.
Answer: x = 2 and -2 in the reals, plus 2i and -2i over the complex numbers. Factor x^4 - 16 as (x^2-4)(x^2+4), then split x^2 - 4 into (x-2)(x+2).
The outbound speed is 50 km/h and the return speed 60 km/h. Convert 4 h 24 min to 4.4 h, write both times as distance over speed, and solve 220/v = 4.4.
The answer is 3037. The characteristic equation (r - 2)^2 = 0 gives a_n = (A + Bn)2^n, so a_2025 = 6,149,925 * 2^2025 and b/c = 6,149,925/2025 = 3037.
Term 19 is 91.66672, so its integer part is 91. The gaps halve and alternate in sign, and the sequence converges to (a1 + 2a2)/3 = 275/3, not to 2a2 - a1.
The real solutions are x ≈ ±1.23561. Clearing the decimal gives x⁶ + 25x⁴ + 25x² − 100 = 0, and y = x² turns it into a cubic whose only positive root is y ≈ 1.526734.
Four real roots: x ≈ 0.00096368, 0.14670508, 0.15345189, 0.22484227, plus a complex pair. Bracketing by sign changes is what makes the near-double roots separable.
Answer: x = 2.982928 and x = -3.152475. Expanding gives the sextic x^6+104x^4+40x^3-10000 = 0, which has no rational root, so bracket and bisect both sign changes.
Answer: the system is inconsistent - no solution exists. Substitution reduces it to three equations in y and z that give y = 25/2, z = -9/2, values which fail the remaining equation.
The inequality is equivalent to (k - x)(t - y) > 0, so k - x and t - y must share a sign. Grouping turns four scattered products into one factored condition.
The cubic is irreducible over the rationals. Only 1 and -1 are candidate roots and both fail, so no linear rational factor exists and grouping cannot work either.
No largest m exists: substituting y = 1 − 3x gives 1/x − 3 + 1/(1 − 3x), which tends to +∞ as x → 1/3⁻, so the expression is unbounded above and the bound m ≥ value fails.
Answer: [2, 5] union {10}. Even powers never change the sign, so only (x-2)^3 and (x-5)^3 matter; the isolated point x = 10 belongs because the product is zero.
The answer is 0.6x + 0.2. This is the weighted-average form: mixing fraction x at 0.8 with the rest at 0.2 gives a result rising linearly in x.
The quotient simplifies to 7y⁵/(4x³). Reduce 28/16 to 7/4, subtract the exponents to get x⁻³ and y⁵, then move the x⁻³ down into the denominator to finish.
The simplified expression is a^2*b - ab^2 + b^2 - 2b. The minus sign in front of the second bracket flips every sign inside before like terms are collected.
The result is sqrt(3)*cbrt(4)/4. Pull 2 out of sqrt12 and 3 out of cbrt54, then multiply by cbrt(4)/cbrt(4) because a cube root needs two extra factors.
Answer: 3a, valid for a not equal to 0. Split off the coefficient, subtract exponents so a^2/a = a^1, and note why the original expression needs a nonzero denominator.
The answer is 12x + 5. Distributing the 4 gives 12x + 20, and only the constants combine, since 20 and −15 are like terms while the 12x term has no partner.
The answer is 4d/a^9. Adding the exponents -10 and 1 gives a^-9, and a negative exponent means the factor belongs in the denominator instead.
The answer is 3x + 12. Distribute the 6 across both terms in the bracket to get 6x + 12, then combine the like terms 6x and -3x into a single 3x.
The result is 3(x - y)^2(x + y). Because y - x = -(x - y), the odd cube flips sign to -3(x - y)^3, after which 3(x - y)^2 is a common factor of both terms.
Answer: 77x^5 y^2. Multiply the constants to 77, then add exponents of like bases: four separate x factors and one x^2 give x^5, and the two y factors give y^2.
The simplified form is 13f + 12. Distributing gives 7f + 28 + 6f − 16, so the f terms add to 13f while the constants 28 and −16 combine to give 12.
The answer is just y. Distributing -3 gives -6y + 3z, so the z terms cancel completely and only 7y - 6y survives out of the whole expression.
The expression collapses to x² − 16y² = (x − 4y)(x + 4y). The two xy terms cancel exactly, and that cancellation is what leaves a clean difference of squares.
The simplified form is 2a/b. Divide the coefficients 8/4 = 2, then subtract exponents: a^(2-1) = a stays on top while b^(1-2) = b^(-1) moves to the bottom.
The result is 8m^2 + 20mn + 8n^2 = 4(m + 2n)(2m + n). Writing 9(m+n)^2 as (3m+3n)^2 turns the whole expression into a plain difference of two squares.
The result is 2(x+1)(x^3 - 2x^2 - 3x - 2). Factoring x^2 - 9 lets x + 3 cancel, after which (x+1) is a common factor of all four terms.
The expression collapses to 8x² + 12x + 4 = 4(x + 1)(2x + 1). All the cubic terms cancel, so a degree-3 expression simplifies to a quadratic.
The bracket collapses to 4t, so the first fraction is 32t/(1-t^2)^2 and the expression becomes a single fraction over (1-t^2)^2(1+t^2)^2.
Answer: (x-y)^2 (x+y)(x+y-1). Because (y-x)^2 equals (x-y)^2, both terms share (x-y)^2(x+y), and pulling that out leaves the simple bracket (x+y) - 1.
The result is (a + 2b)^2. Expanding gives a^2 - 4ab + 4b^2, and adding 8ab turns the middle term from -4ab into +4ab, flipping the sign inside the square.
The whole expression simplifies to 1. The denominator is the perfect square (root a + root b)^2 and the numerator is a difference of squares in root a and root b.
The expression collapses to 2 - a - 2b. The inner bracket becomes (2a+b)/(a+b), and its reciprocal cancels both (a+b) and (2a+b) at a stroke.
The answer is (a - x^2)/x, valid for x not zero. Rewrite the lone x as x^2/x so both terms share the denominator, then subtract the numerators in one step.
Answer: 6 root3 x^(62/21) / (35 y^(3/2)). Convert every radical to a fractional exponent, gather the constants into 18/(35 root3), then add exponents base by base.
The answer is a⁴b. Factoring ab out of the first bracket exposes the sum-of-cubes identity, and the resulting ab⁴ cancels the trailing subtraction exactly.
The value is 1/2. Factoring 2^8 from the numerator and 2^9 from the denominator leaves 7*2^8 over 7*2^9, so the sevens cancel and the powers give 2^-1.
The answer is m^8 / (256 n^44). Divide inside first to get -4m^-2 n^11, then raise each factor to -4; the negative base to an even power turns positive.
The result is (b² + b√b + 2b − 3√b + 1)/(b + √b − 1). Flip the negative exponent, set x = √b so every power is a polynomial, cancel x + 1, then add 2√b.
The expression simplifies to a⁴b. Factor ab(a + b) out of the first bracket, apply (a+b)(a²−ab+b²) = a³+b³, and the ab⁴ term cancels the ab³ piece.
The answer is 1. Any nonzero quantity raised to the power 0 equals 1, so the whole fraction of negative exponents never has to be simplified at all here.
The whole expression collapses to a^4 b. Factor ab out of the first bracket, apply the sum-of-cubes identity, distribute, and the ab^4 terms cancel exactly.
The quotient simplifies to 3x. Combine the two fifth roots into one radical, reduce 729x⁶/(3x) to 243x⁵, and recognise 243x⁵ as a perfect fifth power.
It becomes (L - R)(L + R)/(H + R), and nothing cancels unless H = L, in which case it reduces to L - R. Distinct letters can never cancel against each other.
The simplified form is -125b^24/a^18. Reduce inside the bracket first to -5a^-6*b^8, then cube each factor and rewrite with positive exponents.
Answer: f(x) = -x - 289/x with x not equal to 0. Split the single fraction into two terms, cancel one factor of x, and state the domain restriction.
The simplified form is −80v + 40, or 40(1 − 2v). Distributing −10 gives −90v + 40, and adding the loose 10v leaves −80v with the constant unchanged.
The expression simplifies to −125b²⁴/a¹⁸. Reduce inside the parentheses first with the quotient rule, then cube the coefficient and multiply every exponent by 3.
The answer is -243 x^10 y^20, not the -27 x^10 y^20 often written. Raise every factor to the fifth power: (-3)^5 = -243, and multiply the exponents 2 and 4 by 5.
The simplified form is −9q + 11r. Two of the three groupings carry a negative multiplier, so every term inside them flips sign before like terms are collected.
The whole expression collapses to just a. Combine each bracket over a+b, flip the second one because of the -1 exponent, cancel, then subtract the trailing 3b.
The simplified form is cube root of (7y^2), all over y. Multiply numerator and denominator by the cube root of y^2 so the denominator becomes a perfect cube.
The expression collapses to −4x² + 12x = 4x(3 − x). Changing the last term from 2(x + 1)³ to 2(x − 1)³ changes the answer completely, from 8x² + 12x + 4.
The result is 2|xy| sqrt(5z) / z^3. Split off the even powers, keep absolute values on the odd-rooted variables, then rationalise the leftover sqrt(z) in the denominator.
The simplified form is 9|x| times sqrt(5). Factor 405 as 81 times 5, split the radical, and keep absolute-value bars because sqrt(x^2) equals |x|.
The simplified form is 2i sqrt(35). Pull out the imaginary unit first to get i sqrt(140), then split 140 as 4 times 35 and take the root of 4 outside.
The expression is the constant 64. Setting a = 3x^2+4 and b = 3x^2-4 turns it into (a-b)^2, and a-b = 8, so every power of x disappears from the answer.
The answer is 2x^2 + 3x + 17. The two cubic terms cancel and the difference of squares contributes -x^2 + 16, so only a quadratic survives at the end.
Answer: 1.76566975(x + 1). Group the repeated factors as 1.15^2 and 1.3^2, multiply the constants into a single multiplier, then distribute if a linear form is wanted.
The whole expression reduces to 36x. Expand the two cubes so the x² terms cancel in pairs, handle the −3x product separately, and watch every x³ term disappear.
The result is x^2 - 6. Expanding the product gives x^2 - x - 6, and the loose +x cancels the -x exactly, leaving a binomial with no linear term at all.
The whole expression collapses to the constant 25. Recognise A² + B² − 2AB = (A − B)², subtract the two brackets, and see why every power of x cancels out.
Answer: 0 for every x. Expanding (x+4)^2 gives exactly x^2 + 8x + 16, so the bracket subtracts the same trinomial term by term and everything cancels.
The result is x^2 - 2y. The first product is the difference of squares x^2 - y^2, and the -y^2 cancels against the +y^2 coming from expanding y(y - 2).
The result is x^2/4 - 3x sqrt(5) - 15. Squaring the bracket gives x^2/4 + x sqrt(5) + 5, and tripling then subtracting leaves only a quarter of the x^2 term.
It simplifies to x - 3, valid for x not equal to -3. Factoring x^2 - 9 as (x - 3)(x + 3) exposes the common factor, but the original is undefined at x = -3.
The answer is x^2 + 10. The power 3^2 is a number, not a variable term, so it becomes 9 and then merges with the 1 while x^2 stays separate.
The result is x^(-ab), equivalently 1/x^(ab). The power-of-a-power rule multiplies the exponents, and the negative sign moves the whole thing into a denominator.
The answer is m = −1/3. The rise is +1 and the run is −3, so the negative sign comes from the denominator alone, not from either of the negative coordinates.
The answer is 17. Turn 2 log5(root x + 2) into log5 of a square, drop the logs since base 5 > 1, and the x terms cancel leaving 28 < 7 root x, so x > 16.
x = 87870/0.30 = 292900 exactly. The identical terms combine to 0.30x, and because 878700 is divisible by 3 the answer is a whole number with no rounding.
x = 963. Writing 0.6 as 27/45 and 5/9 as 25/45 shows the two x coefficients differ by only 2/45, so the constant gap of 42.8 is magnified 22.5 times.
x = 52500. Dividing by 0.8 gives 1 + x/10000 = 6.25, subtracting 1 leaves x/10000 = 5.25, and multiplying by 10000 finishes it. Check: 0.8 * 6.25 = 5 exactly.
x = 87870/1.7 = 878700/17 = 51688.24 rounded to two decimals. The two identical terms add up to 1.7x, and the exact quotient 51688 + 4/17 never terminates.
Answer: x = 2,505,589.39. Halve the right side to 1,625,000, multiply 0.85, 1.09 and 0.7 into the single factor 0.64855, then divide to recover x.
Answer: x = 2,063,426.56. Reduce the right side to 1,625,000, collapse the three multipliers into 0.787525, then divide - a two-discount, one-markup chain.
The answer is x = 2500/39, about 64.103. Dividing by 0.78 is the same as multiplying by 50/39, which keeps the answer exact instead of a rounded decimal.
The answer is x = 100. Dividing by 0.8 first gives x + 100 = 200 in one step, which is faster and cleaner than distributing the 0.8 across the bracket.
The solution is x = 325. Add 100 to both sides to get 0.8x = 260, then divide both sides by 0.8 — or equivalently multiply by 5/4, since 0.8 is the fraction 4/5.
The answer is X = 1900. Add 1200 first to get 0.8X = 1520, then divide - and dividing by 0.8 is the same as multiplying by 1.25, so the value grows.
The solution is x = 800. Adding 400 gives 0.8x = 640, and since 0.8 equals 4/5, multiplying 640 by 5/4 produces the answer with no decimal division.
Answer: C = 1.35 + 0.15B and B = (C - 1.35)/0.15. Add 0.15B to isolate C; to get B instead, move C across, divide by -0.15, and watch the sign.
Answer: X = 225. Expand 1.8(X + 100) into 1.8X + 180, combine like terms to reach 0.8X - 20 = 160, add 20 to both sides, then divide by 0.8 to finish.
The solution is x > -2. Multiplying by 6 turns the inequality into 6 - 3(x + 6) < 2(2x + 1), which simplifies to -14 < 7x after collecting terms.
x = -175.86. Because 2.7421 is larger than 2.7, the reciprocal difference is negative, so 1/x = -0.0421/7.40367 and x itself must come out negative.
x = 16.2196, not 16.2214. Isolating 1/x gives (3.3 - 2.7421)/(2.7421 x 3.3) = 0.5579/9.04893, and inverting that single quotient finishes the problem.
x = 17.7272. Isolating 1/x gives (3.3 - 2.7821)/(2.7821 x 3.3) = 0.5179/9.18093, and taking the reciprocal of that fraction gives the answer directly.
x = 1. When two unit fractions are equal and neither denominator is zero, the denominators must match, so 3x + 11 = 9x + 5 and both sides become 14.
The solution is y = x/(8x - 1), with x not 0 and x not 1/8. Combining the fractions gives x + y = 8xy, and collecting the y terms lets y be factored out.
x > 2, that is the interval (2, infinity). Multiplying by the positive 3 gives 1 + x < 3x - 3, and collecting terms leaves 4 < 2x with the direction unchanged.
The real solutions are x = 0 and x = 0.0156549. Spotting the visible root x = 0 and locating the single minimum at x = 0.0079085 proves there are exactly two.
The solution is x = 0.9. Multiplying by 1 changes nothing, so the equation is simply 1 - x = 0.1, which is linear and has exactly one solution.
The solution is x < 5. Moving the x terms to the right keeps the coefficient positive, so the final division by 2 leaves the inequality direction untouched.
x = 104022.99. Distributing the 13% bracket first turns the equation into 139732000 = 1343.28x, a single division once every x-term is collected on one side.
x = 103254.32. With only 2.72x loose on the left, the collected coefficient is 1353.28, and one division finishes the problem.
Answer: x = 12,110.83. Evaluate 12812 x 1.0398 = 13,321.9176, clear the denominator to get 13,321.9176 = 1.1x, then divide by 1.1 to strip the 10% margin.
The solution is x = 13. Add the numerator to get 169, multiply both sides by x to reach 13x = 169, then divide by 13 since 169 = 13 squared.
The answer is x = 96/7, about 13.714. Adding gives 192/x = 14, and 192/14 then has to be reduced by the common factor 2 to reach lowest terms.
The answer is x = 6960. Add the numerator first to get 348/x = 0.05, then cross-multiply - dividing by 0.05 is the same as multiplying by 20.
The answer is x = 317/7.8 = 1585/39, about 40.64. Add the numerator first, then cross-multiply - and note the decimal answer never terminates.
Answer: x = 3200/3, about 1066.67. Rewrite 1.5 as 3/2 and multiply 1600 by the reciprocal 2/3; the decimal repeats, so the fraction is the exact answer.
Answer: x = 202.31 (exactly 176.0096/0.87). Simplify the constant to 176.0096, collect the x-terms into 0.87x, then divide to recover the gross total.
Answer: x = 202,309,885.06. Evaluate the constant as 176,009,600, subtract 0.13x from both sides to get 0.87x, then divide by 0.87 to gross the base up.
x = 16939/1.923 = 8808.63. Multiplying by x and adding x turns the equation into 16939 = 1.923x, which is the rule x = total / (1 + ratio) in disguise.
The answer is [-4, -3) union (1, 2]. The outer inequality gives two rays and the inner one a band, and the intersection is two short intervals.
x = -52/5, or -10.4. Multiplying by x + 10 gives 2 = -5x - 50, and the root is checked against the excluded value x = -10 before being accepted.
The solution is x = -3. The left side is pure arithmetic: 5 - 8 = -3 and 2 + (-3) = -1, so -1 = 2 + x, and subtracting 2 from both sides gives x = -3.
The answer is x is about 1.71562. Mixing an exponential with a linear term admits no closed form, so bracket the root between 1 and 2 and bisect to converge.
The solution is x ≤ 0 or x ≥ 2 with x ≠ 3. Substituting t = 2ˣ turns the second denominator into the perfect square (t − 8)², and the inequality into (u + 4)(u + 7) ≥ 0.
The answer is x = 10. Expanding gives 320 - 20x = 240 - 12x, so 80 = 8x; removing 10 from each bracket makes the two weighted amounts balance at 120.
The answer is x = 5/7. Removing the bracket turns -(4x + 2) into -4x - 2, giving 16x - 4 = 6 + 2x, so 14x = 10 and the fraction 10/14 reduces to 5/7.
The answer is x = 10/3, about 3.33. Combining gives 30x = 100, and 100/30 must be reduced by the common factor 10 rather than left as it is.
The solution is x = 121/2, exactly 60.5. Combine like terms to get 30x = 1815, then cancel the common factor 15 from the numerator and denominator.
The solution is x = 200/3, about 66.67. Combining like terms gives 30x = 2000, and 2000/30 reduces to 200/3, which has no exact decimal form.
The solutions are x = (-34 + sqrt 2941)/21 and x = (-34 - sqrt 2941)/21. No integer pair multiplies to -1785 and adds to 68, so the quadratic formula is required.
Answer: X = -10.26. Simplify the coefficient to 500/27, subtract 2210 to get 190 = -(500/27)X, and the negative sign survives because 2400 exceeds 2210.
The answer is x = 16. Add the numerator to get 400/x = 25, then cross-multiply - the answer counts how many whole parts of size 25 fit into 400.
The answer is t = 20. Subtracting 100t leaves 150t = 3000, so the faster rate needs 20 units of time to overcome a fixed head start of 3000.
Answer: x = 5120/3, about 1706.67. Since 1.5 = 3/2, dividing by 1.5 is the same as multiplying by 2/3, which keeps the answer as an exact repeating decimal.
The solutions are x = (-1 + sqrt 7)/5 and x = (-1 - sqrt 7)/5. The discriminant is 700, and pulling out sqrt(100) turns 10 sqrt 7 over 50 into the reduced form.
Answer: x = 2/3, a triple root. The left side is exactly (3x - 2)^3, since 27x^3 = (3x)^3 and 8 = 2^3 with the middle terms matching -3a^2b and +3ab^2.
Answer: x = 216.809. The fraction 0.05/0.95 reduces to exactly 1/19, so the equation collapses to 228.22 = (20/19)x, giving x = 228.22 times 19/20.
x = 298/13 = 22.9231. Collecting terms gives 1.3x = 29.8, and clearing the decimals shows the exact value is 298/13, which is a non-terminating decimal.
The answer is x ≈ 281.46. Collecting the x terms gives 1.01543594x = 285.8, because the fee fraction 0.0152/0.9848 is just another coefficient of x.
Answer: x = 294.9308, not the 111.241 sometimes quoted. The two 1.52% factors multiply, so the coefficient is only 0.00023461 and x barely moves from 295.
x = 2981/5.3043 = 562.00. Adding x to both sides gives 2981 = 5.3043x, so x = total / (1 + ratio); the leftover 2419.00 is then 4.3043 times x exactly.
The answers are x = 1 and x = 2. A line can cross a convex exponential curve at most twice, so once both integer roots are found the search is provably complete.
The roots are x = 1 and x = 1/3. Recognise (2x − 1)² − x² as the difference of squares a² − b², factor into (x − 1)(3x − 1), then use the zero-product property.
x = 1/2 or x = -7/3. The equation is already factored, so set each bracket to zero separately and solve the two linear equations that result.
The solution is x <= 4, or (-inf, 4]. Add 3 to undo the subtraction, then divide by the positive 2, which leaves the inequality direction unchanged.
The solutions are x = 2 and x = 4/3. Both terms are already squares, so factoring as (A − B)(A + B) gives the two linear factors x − 2 and 3x − 4 directly.
The answer is x < 51/28, about 1.8214. Multiplying every term by the LCM 12 removes both fractions at once, turning it into 24x - 15 < 36 - 4x.
The solution is x > -4. After expanding, the x terms collect on the right to give -4 < x, which is then simply reread with the variable written first.
The solution is x < 7/10. Collect the x terms on the left and the fractions on the right, then add 1/2 and 1/5 over their common denominator of 10.
The solution is x <= 0. The bracket is preceded by a minus sign, so both inner terms change sign, giving 2x - x + 1, which collapses to x + 1 <= 1.
The answer is x > 5. Subtract x to leave x + 1 > 6, then subtract 1; the boundary x = 5 makes both sides equal 11, so it is correctly excluded.
The answer is x > 6. Subtract x from both sides to leave x + 1 > 7, then subtract 1; because no negative divisor appears, the sign never flips.
The answer is x = 4. Subtract 2 from both sides to get 2x = 8, then divide by 2; undoing the addition before the multiplication is what keeps it simple.
The solution is x = -mu / (2(1 + lambda)) when lambda is not -1. At lambda = -1 the x terms cancel, leaving no solution unless mu is 0, when every x works.
The solution is x = −μ/(2(1 + λ)) whenever λ ≠ −1. If λ = −1 the x terms vanish: every real x works when μ = 0, and there is no solution when μ ≠ 0.
x = 2. Expanding (2x + 3)^2 makes the 4x^2 terms cancel, so the quadratic-looking equation collapses to the linear equation 12x + 9 = 51 - 9x.
The solution is x = 2, y = 1. Solve the second equation for x = y + 1, substitute into the first to get 5y + 2 = 7, then back-substitute to recover x.
The solution is x = 4. Subtracting 5 from both sides gives 2x = 8, dividing by 2 gives x = 4, and substituting back returns 13 exactly as required.
The answers are x = (1 + root 3)/2 and (1 - root 3)/2. The discriminant 12 is not a perfect square, so the roots are irrational and factoring cannot work.
The solutions are x = 1 and x = 3/2. With a leading coefficient of 2, split -5x using factors of 2 times 3 = 6 that sum to -5, namely -2 and -3.
The solution is x < -1 or x > 1/2. Factoring 2x^2 + x - 1 as (2x - 1)(x + 1) exposes the two roots, and an upward parabola is positive outside them.
The answer is y <= (40 - 3.8x)/7.7, roughly y <= 5.19 - 0.49x. Dividing by the positive 7.7 keeps the direction, so the region sits below the boundary line.
The answer is x > log base 4 of 3 x 10^9, about 15.74, so the least integer is 16. Since 4^x is always positive, no inequality sign ever flips.
The answer is x = -10/7, or about -1.4286. Distributing the -2 over (2 - 5x) gives -4 + 10x, so the equation becomes 3x - 9 = 1 + 10x, hence -10 = 7x.
The answers are x = 27 exactly and x is about 1.15082. Taking logs reduces it to ln(x)/x = ln(3)/9, a curve that meets a horizontal line twice.
The solution is x < 0.0000020000004 or x > 14.3704368. The convex function 3^x - 500001x dips below zero once, so the line is crossed at exactly two places.
The answer is x = 5 + 3^(49/32), about 10.3776. Isolate the log first, convert to exponential form, then confirm x > 5 so the logarithm is defined.
The answer is x = 5.455/32.63, about 0.16718. Subtract the constant first, then divide - two operations, and the decimals never need clearing.
The answer is x = 10.455/32.63 = 0.32041, which rounds to 0.3204 - not 0.3203. The fifth decimal digit is a 1, so the fourth place rounds down.
The solution is X = 27. Cross-multiplying gives 3150 + 54X = 7200 - 96X, so 150X = 4050; both sides of the original equation then equal 32.
The solution is -1 < k < 1. Dividing by the positive 36 leaves 1 - k^2 > 0, and k^2 < 1 means k lies strictly within one unit of zero on the number line.
x = 1800 + 900 sqrt 7 = 4181.18 or x = 1800 - 900 sqrt 7 = -581.18. The discriminant 22680000 equals (1800 sqrt 7)^2, so both roots simplify neatly.
The solutions are a = -3 and a = -5. Move -45 across, divide the whole equation by the common factor 3, then factor a^2 + 8a + 15 as (a + 3)(a + 5).
The single solution is x = 4/9. Both products contain 9x², so expanding makes the quadratic terms cancel and what looks like a quadratic is really a linear equation.
Answer: no integer solution; the real roots are x = 8 + or - (14)^(1/4)/3, about 8.6448 and 7.3552. Dividing by 7^3 leaves (3x - 24)^4 = 14, which is not a fourth power.
The solution is x >= 1. Adding the coefficients 3 and -2 leaves a single x, so the inequality reduces to x - 1 >= 0 and needs only one more step.
The solution is x > 1. Distribute the 2 over x - 2 to get 2x - 4, subtract 2x from both sides, and the single remaining x needs no division at all.
Answer: x = 3, a double root. With u = 3x - 5 the equation is u^2 - 8u + 16 = (u-4)^2 = 0, so u = 4 and 3x = 9.
The solutions are x = (5 + sqrt 43)/3 and x = (5 - sqrt 43)/3. Multiplying through by x turns the rational equation into the quadratic 3x^2 - 10x - 6 = 0.
The solution is x > 15. Subtracting 2x leaves a coefficient of exactly 1, so after moving the constant 15 across the answer is read off directly.
The answer is x < 4. Multiplying through by 4 gives 3x + 4 > 8x - 16, and collecting on the right avoids ever dividing by a negative number.
The answer is x = 10, and not 25/3. The right side is 35, so 3x = 30 - the equal constants on both sides cancel and never enter the division.
The answer is x = -7, the only solution. The right side can never be positive, so 3x + 5 <= 0 forces x <= -5/3 and only one case has to be checked.
The solution is x = 5. Subtract the 7 that was added last, leaving 3x = 15, then divide by the 3 that multiplied x first, and substitute back to check.
The answer is x < 2/3. Subtract 3x so the x coefficient stays positive, subtract 7, then divide by 3 — no sign flip is needed at any stage this way.
The answer is x < 3/4. Subtract 3x so the coefficient stays positive, subtract 6 to get 3 > 4x, then divide by 4 and read the inequality backwards.
The answers are x = 4 and x = -4. With no middle term you can isolate x^2 = 16 and take roots directly, but the plus-or-minus sign is mandatory.
x = 0 or x = 1/3. There is no constant term, so x is a common factor; dividing both sides by x instead would silently destroy the root x = 0.
The answers are x = (-5 + root 97)/6 and (-5 - root 97)/6. Since 97 is prime and not a perfect square, the roots are irrational and cannot be factored.
The roots are x ≈ −3.1660 and x ≈ 2.9426. Their sum −0.2234 equals −b/a and their product −9.3155 equals c/a, which validates both decimal answers at once.
The roots are x ≈ −2.6457 and x ≈ 2.5303. Vertex form is −4.4912(x + 0.05769)² + 30.0809, so x + 0.05769 = ±√(30.0809/4.4912) = ±2.5880 gives both roots.
The roots are x ≈ −2.1144 and x ≈ 2.0515. Multiplying through by −1 gives 4.6173x² + 0.2906x − 20.029 = 0, which keeps every sign in the formula positive.
The solution is m < 7/3. Group the constants 4 and 3 into 7 and the variable terms -2m and -m into -3m, then move -3m across to avoid a sign flip.
The solution is x > −7. On the right the +6x and −6x cancel, leaving the constant −12; dividing by −4 at the end reverses the inequality sign.
The solution is x = log base 4 of 80, about 3.1610. Since 80 is not a power of 4, taking logarithms and dividing ln 80 by ln 4 gives the exact and decimal answers.
x = 3/10 or x = -11/4. The discriminant 14884 equals 122^2, a perfect square, so the roots are rational and 40x^2 + 98x - 33 factors as (10x - 3)(4x + 11).
The answer is x = 125/7, about 17.857. Dividing gives 750/42, and the two share a factor of 6, so the fraction reduces exactly once and then stops.
x = 4402/73, about 60.3014. Distributing the minus sign gives 3.65x + 0.35 = 220.45, so 3.65x = 220.10, and the quotient is not the round number 60.3.
x = 4405/73, about 60.3425. After distributing, the equation reads 3.65x + 0.35 = 220.6, so 3.65x = 220.25 and dividing gives a non-terminating decimal.
x = 4910/73, about 67.2603. The bracket expands to -0.35x + 0.35, leaving 3.65x = 245.50, and 73 being prime keeps the answer a non-terminating decimal.
x = 8. Distributing gives -60 + 3x, so the equation becomes 7x - 60 = -4; adding 60 and then dividing by 7 leaves x = 8, and 32 - 36 = -4 confirms it.
The answer is x = -8. Subtract 4x to gather the variable on the right, subtract 9, then divide by 2; both sides then evaluate to -39 as a check.
The solution is x < 2. Adding 2x to both sides gives 6x + 8 < 20, then 6x < 12, so dividing by the positive 6 keeps the inequality direction.
The solution is every real number except x = 1/2. The left side factors as the perfect square (2x - 1)^2, which vanishes only at its repeated root x = 1/2.
The solutions are x = (1 + sqrt 33)/8 and x = (1 - sqrt 33)/8. Since 33 = 3 times 11 is squarefree, the radical cannot be simplified and the fraction stays as is.
The two solutions are x = -3/2 and x = -7/2. The discriminant is 400 - 336 = 64, a perfect square, so both roots come out as exact fractions.
The solution is x = 3. Distributing -2 across x - 1 gives -2x + 2, so the equation becomes 7 - 2x = 1, then -2x = -6, and finally x = -6 / -2 = 3.
The solution is x <= -2. Distributing gives 3 - x/2 >= 4, hence -x/2 >= 1, and multiplying both sides by -2 reverses it, producing x <= -2 not x >= -2.
The answer is x = (5ln3 - 2ln5)/ln15, about 0.8398 - and not 1.037. Different bases force logarithms, and ln5 + ln3 collapses neatly into a single ln15.
x = 5399/5.8116 = 929.00. Clearing the fraction gives 5399 = 5.8116x, so x = total / (1 + ratio); the leftover 4470.00 is then exactly 4.8116 times x.
The solution is x = 5. There is no constant term to move, so dividing both sides by 580 finishes the problem, and 2900/580 reduces exactly to 5.
The solution is a < -2 or a > 6/5. The discriminant 256 is a perfect square, giving the roots -2 and 6/5, and the upward parabola is positive outside them.
The answer is w = -7z/2. Collect the w terms on the left and the z terms on the right to reach 2w = -7z, then divide by 2 without touching the sign.
The solution is x = 5. Subtracting 2x leaves 3x - 4 = 11, adding 4 gives 3x = 15, and dividing by 3 yields x = 5, which both sides confirm as 21.
The solutions are x = −2 and x = 1/3. Recognising 49x² as (7x)² turns the equation into A² − B² = 0, which factors into two linear equations instead of a quadratic.
The solution is x > -5/4. Combining 5x and -2x gives 3x, and dividing the fraction -15/4 by 3 reduces to -5/4 with the inequality direction preserved.
The solutions are x = 2 and x = -2. With no linear term you can isolate x^2 = 4 and take roots directly, which is faster than the quadratic formula here.
The answer is x = 280 exactly. The rate gap is only 0.05, and 14/0.05 is a whole number because dividing by 0.05 is the same as multiplying by 20.
The answer is x = 14/0.17, about 82.35. The two x terms nearly cancel, leaving only 0.17x - which is why a small constant produces a large answer.
The answer is p = -7. Expanding both sides gives 30 + 6p = 7p + 37, and subtracting 6p leaves p on its own with no division step needed at all.
The three real roots are about -0.91311, about 1.11768, and exactly 36, because 36^18 = (6^2)^18 = 6^36. The function ln(x)/x controls the two positive roots.
Answer: X = 4347/50 = 86.94. Combine the two fractions into 500/27, subtract 2210, and this time the right-hand side is smaller so X comes out positive.
The answer is g = 18. The three g terms collapse to 2g because the bare g counts as 1g, so the equation becomes 2g = 36 before any division.
The solution is p < (3 - sqrt 21)/6, or 0 < p < 1/2, or 1 < p < (3 + sqrt 21)/6, because the quintic factors neatly as p(p - 1)(2p - 1)(3p^2 - 3p - 1).
Answers: s = -6 and s = -64/7. Substituting u = s + 9 turns the equation into 7u^2 - 20u - 3 = 0, which factors as (7u+1)(u-3) = 0.
The answer is k = 34.5. Subtracting 300 first gives 200k = 6900, and since 6900 is not a whole multiple of 200 the answer lands on an exact half.
The discriminant of x^2 - 37x + 345 is -11, so there is no real answer. The maximum of 74x - 2x^2 is 684.5, which falls just short of the 690 target.
The answer is x >= 7. Subtract 4x, add 15, then divide by the positive 3 - because the divisor is positive, the inequality sign never has to flip.
The solution is x > -2. Distribute the 3 across the bracket, collect the x terms on the left and the constants on the right, then divide by the positive 4.
The solution is x = -5. Subtracting 3x collects the variable on the left as 4x, then moving the 8 across gives 4x = -20 and dividing by 4 finishes it.
Answer: x = -0.0058737255. Adding 10000x to both sides gathers the coefficient into 1020000, and the constant gap of -5991.2 makes x small and negative.
Answer: x = 7300/13, about 561.54. Distribute both brackets, collect 29200 - 27x, move the x-terms to one side to get 52x = 29200, then divide.
The solution is x = 1.55095. Because the same number multiplies x and stands alone, factoring gives x + 1 = 215900.9/84635.579, one division instead of two steps.
g > 5. Distributing gives 2g - 6 < 8g - 36, and moving the g terms to the right keeps the coefficient positive, so 30 < 6g with no sign flip needed.
The answers are x = 0 and x = 2. Move everything to one side and factor out 8x; dividing both sides by x instead would silently destroy the root x = 0.
The answer is a - b = -4. The individual values of a and b cannot be found from one equation, but their difference is pinned down by subtracting 9 from both sides.
x = log base 9 of 252 = ln 252 / ln 9 = 2.5166. Multiplying by 12 isolates 9^x = 252, which is not a power of 9, so a logarithm is unavoidable here.
x = 9127/1.934 = 4719.23. Multiplying by x and then adding x gives 9127 = 1.934x, which is the rule x = total/(1 + ratio) with a ratio just under 1.
Answer: x = 7517/7, about 1073.86. Divide both sides by 100, isolate the fraction to get 0.08 = 3/(0.14(x - 806)), then invert it and add 806 back on.
x ≈ 1.2503. Both bracketed products share the factor 4 − 2x, whose combined coefficient is 41.4 + 28 = 69.4, reducing everything to 305.6 − 122.8x = 152.0621.
Answer: Vout = 4.81748. The linear coefficients 96.844679 and 97.109892 almost cancel, so every digit matters; exact fractions give -0.265213 V - 3.6354e-7 V^3 + 1.2777 = 0.
The roots are -0.01062039, 0.02187147 and 245.69974892. Two roots are near zero and one is huge, so Vieta checks alone cannot confirm the small ones.
Answer: y > 3/8. Clear the decimal denominators 0.03, 0.5 and 0.15 by rewriting each quotient as a fraction, combine like terms, then divide by 8 to get y > 3/8.
The real root is x = 0.0536118. Bracketing must use correctly evaluated test points: f(0.06) is +0.0246, so the sign change lies below 0.06, not above it.
Answer: X = 14487500/1313781, about 11.027. Clear the products 3.6 x 7 x 60 = 1512 and 25.2, collect the X terms, then divide 11.498016 by 1.0426833.
The roots are x = 15.97929 and x = 8.02999. The discriminant is 0.5945668496; a single mistyped digit in 4ac shifts both roots in the third decimal place.
The roots are x = 1 (double), -1, -5 and plus or minus i sqrt 2. A rational root, then grouping, then a substitution y = x^2 break the sextic apart.
Clearing decimals gives y^3 + 26y^2 + 25y - 100 = 0 with y = x^2. Its one positive root is y = 1.50569, so x = +/- 1.22706 are the only real solutions.
The answer is x is about 5.666. Peel the logarithms one at a time by exponentiating base 10 twice, after collapsing the right-hand side to a single number.
The roots are x ≈ −1.4580 and x ≈ 1.4395. The discriminant is 191.131, and its square root 13.8250 dominates the tiny linear coefficient, so the roots are nearly symmetric.
The solution is x = -1309/110 = -11.9. Evaluate the four triple products first, combine the constants to 1309, then divide by the coefficient 110.
x = 193.92W/0.87 = 222.90W. The unknown appears on both sides, so the 0.13x must be collected before dividing - it is a linear equation, not an evaluation.
The solution is q = 5. Combining the two constants on the left into -20 first, then subtracting 4q from both sides, turns the equation into 6q = 30.
Answer: x = 591.4491. Because the same coefficient appears twice you can factor it out, giving x + 1 = 215900.9/364.4210 and a one-line answer of x = 591.4491179.
The solution is x = 6710/0.97 = 6917.53. Collapsing the bracket to x + 1000 first turns the whole equation into 0.97x - 210 = 6500 in a single step.
x = 2. Expanding all four brackets gives 200 - 56x = 18x + 52, so 148 = 74x. Both sides of the original equation equal 88 at x = 2.
The solution is x = 12/5 and y = root 6 over 5. Scaling the equations by root 2 and root 3 turns both y-coefficients into root 6 so they cancel on addition.
x = 37.8 exactly. Substituting u = x/54.9 collapses the equation to 76.25u + 152.5 − 152.5u = 100, so u = 0.6885 and x = 54.9 × 52.5 / 76.25.
The answer is x = plus or minus root a when a > 0. Multiplying by x gives x^2 = a, but x = 0 is barred by the original fraction, so a = 0 has no solution.
x = (2880 - m)/9 and y = (1440 - 2m)/9, and the inequality x + y <= 8m/3 holds exactly when m >= 160. One sign slip when isolating x reverses the whole condition.
The solution is x = 2201834.869. Turning 3% and 13% into 0.03 and 0.13 makes the x terms combine to +0.10x, since -0.03x + 0.13x leaves a tenth.
The solution is x <= 47.94, or x <= 47 for whole numbers. Dividing by the negative coefficient -0.85 is what flips the inequality sign.
Answer: x = 2000.8486. Evaluate 0.35 x 1077.38 = 377.083 first, subtract it, then divide by 0.35; equivalently x = 1077.38 x 0.65/0.35 in a single line.
The solution is a = 12. Adding -16 is the same as subtracting 16, so the equation becomes a - 16 = -4, and adding 16 to both sides isolates a.
The solution is x = -385.8956. Combine the numerator to 1202.61 - x, cross-multiply to 9(1202.61 - x) = 16(1279.43 + x), then collect the x terms.
The solution is x >= 1/2. Expanding both sides gives x^2 + 4x + 3 >= x^2 - 6x + 8, and subtracting x^2 leaves an ordinary linear inequality.
Factoring out z leaves the depressed cubic z^3 + 2r^2 z - 8 = 0 with r^2 = x^2 + y^2, whose single real root Cardano formula gives explicitly.
The solutions are x = 0.9 exactly and x = 0.8974433. Factoring (8.9-x)^2 - 64 as (0.9-x)(16.9-x) exposes a common factor and reduces a quartic to a cubic.
The answer is x ≈ 4.9368×10⁻⁵. Evaluate the known reciprocal cube as 8×10¹², add it across, then invert and take a cube root — all in scientific notation.
The only real solution is s = -3.0864085. Clearing denominators gives 2s^3 + 33s^2 + 180s + 300 = 0, which has no rational root despite appearances.
The solution is a = −1/550, b = 199/1100, c = 61/55. Because c has coefficient 1 in all three equations, subtracting consecutive pairs removes it in one stroke.
Answer: x1 = 19, x2 = -8, x3 = 1. Swap rows to get a leading 1, clear below the pivots, then back-substitute to reduced row echelon form and read off the solution.
D = 12830.69 and W = 12169.31. Substituting W = 25000 - D collapses the second equation to 0.945D = 12125, since the two rates differ by 0.945.
The solution is X = 4963/11 ≈ 451.18 and Y = 12142/11 ≈ 1103.82. Multiplying the second equation by 5.15 and subtracting isolates Y in a single step.
The solution is X = 2393/18 ≈ 132.94 and Y = 3218/9 ≈ 357.56. Substituting X = 490.5 − Y turns the system into one equation in Y with coefficient 0.45.
The solution is X = 4481/11 ≈ 407.36 and Y = 13064/11 ≈ 1187.64. The coefficient determinant is 5.20 − 5.75 = −0.55, which drives both quotients.
The solution is X = 524.625 and Y = 1103.375. The average price is 6627.75/1628 ≈ 4.0711, and its position between 3.80 and 4.20 fixes the split directly.
The solution is x = -339.79. The x coefficients differ by only 0.14394, so the constant gap of -48.9091 is magnified about seven times when you divide.
The solution is x = -374.70. Subtracting the smaller x term leaves a coefficient of only 0.12987, so a small constant difference produces a large negative root.
The answer is exactly x = 37.8, not just an approximation: 52.5/76.25 reduces to 42/61 and 549 = 9 × 61, so 54.9 · 42/61 collapses to 378/10 with no rounding.
The answers are x = -6 and x = 10. Simplify the right side to 8 first, then split into 2 - x = 8 and 2 - x = -8 to catch both of the solutions.
The answer is the closed interval -7/2 <= a <= 1/2, not isolated points. Between the two kinks the a terms cancel and the equation becomes 8 = 8.
The answer is x = 3/4 only. The right side must be non-negative, so x >= 0, and that constraint eliminates the second case's candidate x = -3/8.
The solution is -5/2 < x < 4. The breakpoints -1/3 and 4 give three cases, and the last one collapses to x + 1 < x + 1, which is never true anywhere.
The solutions are x = 16/3 and x = −4/5. When two absolute values are equal the insides are equal or opposite, so split into 4x − 6 = x + 10 and 4x − 6 = −x − 10.
x = 4/7 or x = -2/7. Because 3 is positive the equation splits into 7x - 1 = 3 and 7x - 1 = -3, so the two roots land 3/7 on either side of x = 1/7 exactly.
The answer is -2 < x < 2. Between the kinks the sum is constantly 2, and outside it grows by 2 per unit, so it reaches 4 exactly one unit past each kink.
The answer is x < 0 or x > 7. Between the kinks the sum is constantly 3, so the total only reaches 7 two units beyond each of the two kink points.
The answer is x < -3 or x > 0. The middle interval collapses to the false statement 1 > 3, so the solution set is the two outer rays and nothing between.
The answer is x = 11. The equation says x is equidistant from -66 and 88, so it must be their midpoint - one line instead of two algebraic cases.
The solution is a = 78.77. Taking 13% of 10% means multiplying by 0.013, so the deduction is only 0.091a and the a coefficient stays positive at 6.909.
The solutions are b = 4 + sqrt 11 and b = 4 - sqrt 11. Rename the formula's coefficients to A, B, C so the variable b is never confused with the middle coefficient.
The solution is x = 2034022.99. Moving 0.13x across leaves 0.87x = 1769600, because the right-hand x has an unwritten coefficient of exactly 1.
x = m, provided G and d are nonzero. Both sides carry the same denominator d^2 and the same factor G, so cancelling leaves Gm = Gx and therefore x = m.
The solution is x = (4k - 6)/(k - 3) for k not equal to 3. When k = 3 the equation reduces to 6 = 12, a contradiction, so there is no solution at all.
x = exp(5 ln2 ln3/(ln2 + ln3)) = 8.3729, not 6.06. Converting both logarithms to natural logs turns the equation into a single linear equation in ln x.
The exact solution is x = −22703/143 ≈ −158.7622. Subtract 0.06997 to get −0.00143x = 0.22703, then divide by the negative coefficient, which flips the sign of the answer.
x <= (65 - 5 sqrt 161)/2 = 0.7786 or x >= (65 + 5 sqrt 161)/2 = 64.2214. Dividing by -4 reverses the inequality to x^2 - 65x + 50 >= 0, true outside the roots.
x = -3. The bare leading minus turns -(5x - 4) into -5x + 4, and after collecting terms -3x - 10 = -1, so -3x = 9, and dividing by -3 then gives x = -3.
The answer is x < 7. Dividing by the negative -6 reverses the direction, so the greater-than becomes a less-than - the single step that decides the answer.
The answer is x = -2. Simplify the right side to 14 first, then divide by -7 - a positive divided by a negative gives a negative quotient, not 2.
t = -3 + sqrt(37), about 3.0828. Multiplying through by -1 gives t^2 + 6t - 28 = 0 with roots -3 +/- sqrt(37), and the condition t > 0 rules out -3 - sqrt(37).
There is no real solution; the complex roots are m = (1 + i sqrt 3)/2 and (1 - i sqrt 3)/2. Clearing m gives m^2 - m + 1 = 0 with discriminant -3.
The solution is x = 1/(2 - m), valid whenever m is not 2. Isolate the reciprocal, flip both sides, and note that m = 2 makes the denominator vanish.
The solution is 9/2 <= x < 5. Subtracting 8 from all three parts gives -10 < -2x <= -9, and then dividing by -2 reverses both inequality signs at once.
The answer is x = 3. Add 9 to both sides to get -2x = -6, then divide by -2; two negatives divide to a positive, so the answer is positive 3.
The answer is x <= -2 or x >= 2. Dividing by the negative -2 reverses the sign to x^2 >= 4, and a greater-than square gives two outward rays.
The solution is x >= -3. After subtracting 6 you divide by -4, and dividing an inequality by a negative number reverses its direction from <= to >=.
x = 2 or x = 3. Pulling out the common factor -4 leaves the simple monic quadratic x^2 - 5x + 6, which factors as (x - 2)(x - 3).
The answer is x < -4/3. Dividing both sides by -6 reverses the inequality, and the resulting fraction -8/6 reduces to -4/3, or about -1.333.
The solutions are t = -3 + sqrt 37 and t = -3 - sqrt 37. Multiplying by -1 gives t^2 + 6t - 28 = 0, whose discriminant 148 simplifies to 2 sqrt 37.
The answer is x < 1. Moving the 1 across gives -x > -1, and dividing by -1 flips the greater-than into a less-than, which is the whole difficulty here.
The solution is 1 <= x <= 3. Multiplying by -1 reverses the inequality into x^2 - 4x + 3 <= 0, which holds between the roots 1 and 3 inclusive.
The solution is every real number except x = 2. The expression equals -(x - 2)^2, which is negative everywhere apart from the single point where it is zero.
There is no solution. Flipping to x^2 - 6x + 10 < 0 gives a discriminant of -4, so that upward parabola never reaches the axis and is always positive.
The roots are x = 3 and x = -1/2. Multiplying by the LCD of 30 gives 2x^2 = 5x + 3, and factoring 2x^2 - 5x - 3 as (2x + 1)(x - 3) finishes the job.
x < -4, that is the interval (-infinity, -4). Multiplying both sides by the positive number 3 keeps the inequality direction; only a negative multiplier flips it.
The four roots are x = −1, 3/2, 2 − √3 and 2 + √3. Factor the quartic with the rational root theorem and synthetic division, then use the quadratic formula.
Answer: x = 24.5. Expand to 367.5 + 4x = 372.4 + 3.8x; the x terms differ by only 0.2, so a gap of 4.9 in the constants needs 24.5 units of the new material.
The answer is -3 < x < -2. Subtracting 4 from all three parts is the only step needed, and both bounds land negative because 4 exceeds them both.
The roots are x = −3, x = 1/2 and x = 1. The rational root test flags x = 1/2, so 2x − 1 divides out and leaves x² + 2x − 3 = (x + 3)(x − 1).
The answer is x > cube root of 5, about 1.71. A cube root is defined for all reals and is strictly increasing, so the answer is one ray with no case split.
The answers are x = 1 (a double root) and x = -2. Synthetic division by x - 1 leaves x^2 + x - 2, which contains the factor x - 1 a second time.
The real root is x ≈ −0.6535952, with complex roots x ≈ −0.6732024 ± 2.3805031i. No rational root exists, so bracket between −1 and 0 and refine with Newton's method.
The real answer is x = 7, since the cubic is x(x+1)(x+2) = 504 and 7*8*9 = 504. The other two roots are the complex pair -5 plus or minus i root 47.
Answer: x = (-3 +/- sqrt(555))/2, about 10.279 and -13.279. Move 2.91 across, multiply by 100 to get 2x^2 + 6x - 273 = 0, then use the quadratic formula.
The solution is x < 0. Combine -x - x into -2x, multiply both sides by the positive number 3 so the sign is unchanged, and collect the x terms.
The solution is x < 3/2. Combine 2x - x into x, multiply both sides by 3 which keeps the inequality direction, then collect terms to get 6x < 9.
The answer is x < -|a| or x > |a|. You cannot take square roots directly because a may be negative; factoring into (x - a)(x + a) > 0 handles every sign of a at once.
The answer is x > 1. Taking logs turns it into x ln(x) > 0, and since x is positive the sign depends only on ln(x), which is positive exactly above 1.
Answer: x = 40000/59, about 677.97. Distribute 0.21, combine the x terms into 0.59x, subtract 2100 and divide; keeping the fraction avoids rounding drift.
Answer: x = 142.393 or x = 17.619. The discriminant is 426213.1225 - 167051.52 = 259161.6025, whose square root is 509.0792; divide by 2a = 8.16 to finish.
The five roots are x = 0, 10, −5, −15, −20. Factor out x, then split the quartic into (x² + 10x − 200)(x² + 20x + 75) — every root is an integer.
The solutions are x = 0 and x = 1/16. Rewriting the nested radical as x^(3/4) turns the equation into x^(3/4) = 2x, and x = 0 must be handled separately.
x = 3 and y = 3. Substituting y = 6 - x collapses the first equation to 120x + 600 = 960, and a weighted-average check confirms the even split of six units.
The answer is x = 4, y = -5. The first equation gives y = 3 - 2x with no fractions at all, which makes substitution cleaner than elimination here.
The solution is x = 14/5 and y = 9/5. Rearrange x − y = 1 into x = y + 1, substitute into 3x + 2y = 12 to get 5y + 3 = 12, then back-substitute.
Answer: x = 1 and y = 1. Scale the equations by 3 and 2 so the y terms become +6y and -6y, add to eliminate y, and 13x = 13 gives the answer immediately.
The answer is x = 17/4 and y = 3/4. Dividing the second equation by 2 gives x + y = 5, after which subtracting eliminates x in a single step.
The solution is x = 3, y = -2. The y coefficients are already opposites, so adding the two equations eliminates y immediately and leaves 3x = 9 in one step.
The answer is x = 4. Add 3 to both sides to get (3/4)x = 3, then multiply by the reciprocal 4/3 rather than dividing by a fraction, or clear the 4 first.
The solutions are x = -2, 0 and 11. Move everything to one side, factor out the common x, then factor the remaining quadratic as (x - 11)(x + 2).
x = 5, a root of multiplicity three. The coefficients 1, -15, 75, -125 are the binomial pattern of (x - 5)^3, so there is one triple root and nothing else.
The roots are x = 1, 2 and 3. Test the divisors of 6, divide out (x − 1) by synthetic division, factor x² − 5x + 6, then confirm with the Vieta sum and product.
The answer is about 737.56. Collecting the x terms gives 0.82x = 604.8, so the base is the fixed part divided by 1 minus the 18 percent share.
The answer is x = 600/7, about 85.71. Collecting the x terms gives 0.7x = 60, and the denominator 7 is exactly why the decimal never terminates.
The answer is x = 70/0.65 = 1400/13, about 107.69. The unknown sits on both sides, so collect the x terms to get 0.65x = 70 before dividing - not 70/0.35.
The answer is x = 400/3, about 133.33. Moving 0.4x over to the left gives 0.6x = 80, so the self-reference costs a factor of 1 - 0.4, not 0.4.
x = 262962.96. The unknown appears on both sides, so expand the brackets first; the equation collapses to 1.08x = 284000.
Answer: x = 37,500. The unknown sits inside the bracket too, so simplify to x = 0.12(350000 - x), expand to 1.12x = 42000, and divide to finish the problem.
The real roots are x = 1.0524735 and x = -1.6942810. The factored form x(x + 1)(x^2 + 1) guarantees exactly two real roots; the other two are complex.
The real roots are x = 1.0620313 and x = -1.7023038. Bracketing on [1, 1.1] and on [-1.75, -1.7] and then bisecting pins both down to seven decimals.
The two real roots are x = 7.9431476 and x = -8.4477787. Factoring the left side as x(x + 1)(x^2 + 1) shows immediately why exactly two real roots exist.
x = 7246.475 exactly. The left side collapses to 0.8x, which is the algebra of a 20 percent discount, and dividing by 0.8 is the same as multiplying by 1.25.
The answers are x = 0.4940 and x = -0.3340. Expanding gives x^2 - 0.16x - 0.165 = 0, whose discriminant 0.6856 is not a perfect square at all.
The answers are x = 4.143 and x = -3.983. The discriminant is 0.0256 + 66 = 66.0256, whose square root 8.1256 is nowhere near a round number.
The answer is x = 0.19/0.835 = 0.2275. This is the standard margin equation: a fixed cost of 0.19 leaves a 16.5% margin, so the base is cost over (1 - margin).
The answer is x = 50. Subtracting 0.98x leaves 0.02x = 1, and dividing by 0.02 is the same as multiplying by 50, since 0.02 is one fiftieth.
The answer is x = 100. Subtracting 0.99x leaves only 0.01x = 1, and dividing by the small coefficient 0.01 is the same as multiplying by 100.
Answer: x = 3. Multiply every term by the LCD 10 to get 5(x-1) = 20 - 2(x+2), expand both sides carefully, then collect terms to reach 7x = 21 in one move.
The answers are x = (3 + root 13)/2 and (3 - root 13)/2. Multiplying by x turns it into x^2 - 3x - 1 = 0, and neither root is zero, so neither is extraneous.
The exact answer is x = 512009/6 = 85334.83333…. Subtract 3, divide by 0.6 by multiplying by 5/3, then add 1; the repeating decimal signals a non-terminating exact value.
Answer: x = 168300/23, about 7317.39. The bracket expands to 0.08x - 500, so after distributing the minus sign the equation becomes 0.92x - 500 = 7232.
The solution is x <= 1. Combining x and -2x leaves a single -x, and dividing by -1 at the very last step reverses the inequality from >= into <=, giving x <= 1.
The answer is x > 8. Add 3 to both sides to isolate x; the inequality sign never changes because only addition, not a negative multiplier, is involved.
The solution is x > -3. Multiplying through by the LCD 4 gives x - 3 < 24 - 2(3 - 4x), and the minus in front of the fraction flips both inner signs.
Answer: x = 6500. Expand the bracket to 0.024x - 150, distribute the minus sign, and the equation collapses to 0.976x + 150 = 6494 with an exact whole-number root.
The solution is x <= 1. Distributing -3 across x - 2 gives -3x + 6, and the resulting -2x >= -2 flips direction when both sides are divided by -2.
The answer is x > 15. Add 4 to both sides to undo the subtraction; since no negative multiplication occurs, the greater-than sign is unaffected.
The solution is x > -5/4. Multiplying both sides by 6 gives 2(x - 4) < 3(2x - 1), and the answer stays an exact fraction rather than a decimal.
x = 7/2. Equal squares means the bases are equal or opposite; the 'equal' case gives x = 3.5 and the 'opposite' case collapses to the false statement -4 = -3.
The answer is x > 12. Adding 7 to both sides isolates x, and since addition never changes an inequality direction, the greater-than sign stays exactly as it is.
The rearrangements are x = y + 8 and y = x - 8. One equation with two unknowns has infinitely many solutions, forming a line of slope 1 in the plane.
x = 1.9078533, exactly 17/3 times 0.33668. Cross-multiplying leaves 0.15x = 0.286178, and dividing by 0.15 produces a repeating decimal rather than a clean one.
x = 6.39692, which is exactly 19 times 0.33668. Cross-multiplying gives 0.05x = 0.319846, and the 5 percent remainder is what produces the factor 19.
The answer is x = 970/3, about 323.33. Cross-multiplying gives 3x = 970, and that 3 comes from 100 - 97 - which is why the decimal repeats forever.
The answer is x = 490. Cross-multiplying gives 50x = 49(10 + x), so x = 490 - a 98 percent target needs 49 times as much of x as of the fixed 10.
The answer is x < 2. Clear the fraction by doubling both sides, then collecting gives -x > -2, and multiplying by -1 reverses the direction to x < 2.
x = -20/3, about -6.6667. Multiplying every single term by 4 removes both denominators, giving 9x - 6 = 14 + 12x, and so -20 = 3x after collecting terms.
x = (-5 + sqrt 99545)/2 = 155.2538 or x = (-5 - sqrt 99545)/2 = -160.2538. Clearing the fifths and halves turns the product into x^2 + 5x - 24880 = 0.
The answer is x = 44/3, which is 14 and 2/3, or about 14.667. Multiply both sides by 8 to get 88/6, then reduce by the common factor of 2 to finish.
The answer is y = 100x/99, about 1.0101x. Multiplying by y gives x = 0.99y, and dividing by 0.99 turns the decimal into the exact fraction 100/99.
The answer is x = 2/3. Expanding both cubes gives 2x³ − 3x² + 15x − 7, and the subtracted term removes 2x³ − 3x², leaving the linear equation 15x − 7 = 3.
Answer: x = 13.15893318, exactly 92937321/7062679. Divide both sides by the eight-decimal multiplier, then subtract 1; rounding early shifts the result.
The solution is -1 < x < 4. A product of two factors is negative exactly when the factors have opposite signs, which happens strictly between the roots.
The solution is x > 2. Subtract x and then 6 to reach 4 < 2x, divide by the positive 2, and rewrite 2 < x with the variable on the left as x > 2.
The answer is x < 4. Subtract 2 from both sides to isolate x; subtraction preserves the direction, so the less-than sign stays pointing the same way.
The solution is x < 17/2. Collecting the x terms on the right gives 17 > 2x, and dividing by 2 leaves the exact fraction 17/2 rather than a decimal.
The answer is x = 8. Both cubic terms cancel: the first product is the sum of cubes x³ + 64 and the second is x³ − 25x, leaving the linear equation 25x + 64 = 264.
The solutions are x = (-11 + sqrt(61))/2 and x = (-11 - sqrt(61))/2. Expanding gives x^2 + 11x + 15 = 0, whose discriminant 61 is not a perfect square.
x = 1/28 and y = 1/21. Substituting x = 1/12 - y reduces the second equation to 2/3 + 7y = 1, giving y = 1/21; this is a classic two-rate work problem.
The answers are x = 1 + 3root 2 and x = 1 - 3root 2. The discriminant is 72, and simplifying root 72 to 6root 2 is what makes the fraction cancel neatly.
The solutions are x = plus or minus sqrt(2 sqrt 3), about 1.8612. The right-hand side is irrational, so the answer is a nested radical equal to 2^(1/2) 3^(1/4).
x = 5 or x = -5. Recognising 25 as 5^2 turns the left side into (x - 5)(x + 5), which beats taking square roots because it shows both signs automatically.
The solutions are x = 1 + sqrt 5 and x = 1 - sqrt 5. The discriminant is 20, and simplifying sqrt 20 to 2 sqrt 5 lets the whole fraction cancel down neatly.
The solution is every real number except x = 1. The left side is the perfect square (x - 1)^2, which is strictly positive except at its repeated root x = 1.
The first gives (−∞, −2] ∪ [5, ∞) and the second gives [−1, 1]; their intersection is empty, so no single x satisfies both quadratic inequalities at once.
The solution is -2 <= x <= 5. The factorisation (x - 5)(x + 2) shows the upward parabola dips below the axis only between its roots, endpoints included.
The solution is (−∞, −1) ∪ (5, ∞). Factor into (x − 5)(x + 1), mark the roots −1 and 5 on a number line, then test one point in each of the three intervals.
The answers are x = 1 and x = 3. The numbers -1 and -3 multiply to give 3 and add to give -4, so the quadratic factors cleanly as (x - 1)(x - 3).
The answer is y = (x^2 + 1)/4. Isolate the -4y term, then divide by -4 so both signs flip; the result is an upward parabola with vertex at (0, 1/4).
The solutions are x = 2 and x = 3. A positive constant with a negative middle term forces both factors negative, and -2 times -3 is 6 while -2 plus -3 is -5.
The answers are x = 3 + root 14 and 3 - root 14. The discriminant is 56, and simplifying root 56 to 2root 14 lets the factor of 2 cancel with the denominator.
The solution is x < (7 - sqrt 73)/2 = -0.7720 or x > (7 + sqrt 73)/2 = 7.7720. The discriminant 73 is prime, so both boundaries remain irrational numbers.
The answer splits on |a| > 2: the solution is the open interval between (a - root(a^2-4))/2 and (a + root(a^2-4))/2, and is empty when -2 <= a <= 2.
x = 7 or x = -6. Look for two numbers with product -42 and sum -1; they are -7 and 6, giving the factorisation (x - 7)(x + 6) = 0.
x = (-17 + sqrt(769))/2 = 5.3654 or x = (-17 - sqrt(769))/2 = -22.3654. The discriminant 769 is not a perfect square, so this quadratic has no integer factoring.
The answers are x = -1 + root 2 and -1 - root 2. Rewriting as (x + 1)^2 = 2 is faster than the formula here, because the middle coefficient is even.
The solutions are x = -1 and x = -2. Adding 2 to both sides gives x^2 + 3x + 2 = 0, which factors as (x + 1)(x + 2) because 1 times 2 is 2 and 1 plus 2 is 3.
x = -6 or x = 3. Two numbers with product -18 and sum +3 are 6 and -3, so the quadratic factors as (x + 6)(x - 3) = 0.
x = (-3 +/- i sqrt 11)/2. Subtracting 1 gives x^2 + 3x + 5 = 0, whose discriminant is -11, so there are no real roots at all - only a conjugate pair.
The solutions are x = -7 and x = 3. A negative constant term means the factor pair has opposite signs, and 7 times -3 is -21 while 7 plus -3 is 4.
x = -2 or x = -3. Two numbers that multiply to 6 and add to 5 are 2 and 3, so the quadratic factors, and the zero product property then gives both roots.
x = 14 or x = -15. Moving 210 across gives x^2 + x - 210 = 0, and the pair 15 and -14 multiplies to -210 while adding to 1, which is the x coefficient.
The solution is x <= (-1 - sqrt 5)/2 = -1.6180 or x >= (-1 + sqrt 5)/2 = 0.6180, and those two boundary values are exactly the golden ratio conjugates.
The answer is x = 5180/11 = 470.909091, not the round 470. Dividing 518 by 1.10 never terminates, and 1.1 x 470 = 517 falls a full unit short of 518.
The answer is x = 7/12, about 0.5833. Divide both sides by 60 to get 35/60, then cancel the common factor of 5 from numerator and denominator.
There is no real solution: the discriminant of x^2 - 37x + 345 is -11. The complex answers are (37 plus or minus i root 11)/2, so no rectangle of that area exists.
x = 1828/365 = 5.00822, not exactly 5. Collapsing 75 * 365 / 1000 to 27.375 gives 27.375x = 137.1, while 27.375 * 5 is only 136.875 - short by 0.225.
The answers are x = 15 and x = 35. Dividing through by -5 turns the equation into x^2 - 50x + 525 = 0, which factors neatly as (x - 15)(x - 35).
The four roots are x = -2, -sqrt 2, sqrt 2 and 2. The substitution u = x^2 turns the quartic into u^2 - 6u + 8 = 0, which factors as (u - 2)(u - 4).
The solution is x = 10^25. Setting y = log x turns the equation into y^(3/2) = 125, so y = 125^(2/3) = 25 and x = 10^25.
The decomposition is 1/(x - 2) + 2/(x - 2)^2. Writing x as (x - 2) + 2 lets each piece cancel against the denominator with no simultaneous equations at all.
The product is -19. Writing root(-19) = i root 19 first is essential, because the rule root a times root b = root(ab) fails for negative radicands.
The quartic has just two real roots, x ≈ −78.5149849 and x ≈ 80.4985052. Between them the quartic never rises above about −3992, so no root exists anywhere in [0, 80].
The difference is -3x^3 + 6x^2 + 7x - 18 and its degree is 3. Distribute the minus sign over the second polynomial first, then combine like terms.
The difference is −6x⁴y − 2x³y² + 9x²y³ − 3xy⁴ + y⁵. Learn how to negate every term of the second bracket and which two-variable terms actually combine.
Answer: -4 + 15i. Distribute the minus sign over both parts of the second complex number, then add real parts and imaginary parts separately.
The difference equals −2y times the cube root of 2x. Pull the perfect cubes 8 and y³ out of each radical so both terms become like radicals in cube root of 2x.
The result is -2y times the cube root of 2x. Pull the perfect cubes 8 and 64 out of each radical so both terms carry the same radical part, then subtract.
The answer is −2x² − 4x + 62. The whole difficulty is distributing the minus sign across all three terms of g, which flips 4x to −4x and −60 to +60.
The answer is 0. Each fraction simplifies to -(3a + 2) once you notice the denominators are the negatives of factors in the numerators, so the difference cancels exactly.
Answer: (-x^2 - 4x + 14)/((x+5)(x+4)). Cross-multiply onto the common denominator, expand both products, and distribute the minus sign across all three terms.
The system u3 + u5 - u6 = 6 and u8 + u4 = 52 gives d = 5 and u1 = 1. The even-indexed terms form an AP with difference 10, and their 1010-term sum is 5101510.
Answer: the sum is 6 (m = 0, 2, 4). The system is empty when m >= -1, and x + m - 2 = 2 - x has a non-negative integer root when m is even and m <= 4.
The sum of the roots is -3. The domain x >= -5 throws out x = -6, while the radical factor itself contributes the extra root x = -5, so the roots are 2 and -5.
The sum is -3. The domain x >= -5 rejects the quadratic root x = -6, leaving x = 2 from the quadratic factor and x = -5 from the square-root factor.
The sum is 3. The domain is [-1, 4], so the candidate x = 5 is rejected and only x = -1, x = 0 and x = 4 survive the zero-product test on the two factors.
The real root is x ≈ 0.027167. Substituting y = 1 + x makes the left side a geometric series, y²⁷(y¹⁰ − 1)/(y − 1), which is strictly increasing so the root is unique.
The solution is the interval -3/2 <= x <= 3/2. Splitting at x = -1 and x = 1 turns the sum into -2x, then the constant 2, then 2x; only the outer cases bind.
Rationalising gives x = 2 + √3 and y = 2 − √3, so x + y = 4 and xy = 1; the symmetric identities then give x² + y² = 14, x³ + y³ = 52 and 4x² + 7xy + 4y² = 63.
Dividing gives quotient −x²/2 − x/4 + 27/8 and remainder 21/8. Factor the divisor as −2(x − 1/2) first, divide by x − 1/2, then divide the quotient by −2.
The quotient is 2t² + 2t + 1 and the remainder is −14. The missing t² term must be entered as a 0 placeholder, giving the coefficient row 2, 0, −1, −15.
The quotient is 2t + 1 and the remainder is −14, so 2t² − t − 15 = (t − 1)(2t + 1) − 14. A leading coefficient of 2 is fine; only the divisor must be monic.
The quotient is 2x^2 + 4x + 6 with remainder 5. The divisor x - 1 gives c = 1, and the remainder theorem confirms the 5 independently.
Synthetic division gives quotient x² + 2x + 2 and remainder 7. See how the 0 placeholder for the missing x² term and the bring-down rows work, row by row.
The quotient is x² − 4 with remainder 0, so x − 2 divides the cubic exactly. Set c = 2, bring down 1, then multiply-and-add through the coefficients.
The quotient is x² + 2x and the remainder is 5, so the cubic equals (x − 2)(x² + 2x) + 5. Pad the missing x² term with 0 and use c = 2 in the table.
The quotient is x² − 5x + 6 with remainder 0, so the cubic factors as (x + 1)(x − 2)(x − 3). Note that x + 1 means c = −1, not c = 1, in the table.
The quotient is x² + x + 2 and the remainder is 0. The cubic has no x term, so a placeholder 0 must go in that column before the table is run.
The quotient is x² + 4x − 2 and the remainder is −10, so the cubic equals (x − 1)(x² + 4x − 2) − 10. A nonzero remainder means x − 1 is not a factor.
The solution is x = 300, y = 140. Dividing the equations by 8 and 6 gives x + y = 440 and x + 2y = 580; subtracting isolates y immediately.
Answer: x = 35, y = 5. Expand both brackets to reach 3x - 2y = 95 and 2x + y = 75, solve the second for y, and substitute to get 7x = 245 in one line.
z = 14/13 and the pairs are ((28 ∓ √595)/13, (14 ± √595)/13). The trick is spotting that the middle equation rearranges into 2 × the third equation, which pins z immediately.
One of each costs 6.00. Two white plus one yellow is 5, two white plus three red is 10.50, and three yellow plus two red is 11 - solve it by substitution.
The system is inconsistent. Eliminating b and c makes every a cancel and leaves 2 = 100, a false statement, so no triple (a, b, c) can satisfy all three.
The solution is x ≈ 0.21936, y ≈ −0.14766, z ≈ 0.07169. Substituting z = x + y first turns three equations into two, and reading 25.33333 as 76/3 gives exact fractions.
The row sizes form an AP with u1 = 15, d = 4, n = 30, so the last row holds 131 seats and the total is (30/2)(15 + 131) = 2190 seats.
The combined volume is 16.6xy cubic metres, which is 249 cubic metres at x = 5 and y = 3. Tripling both base dimensions multiplies that base area by 9, not by 3.
The answer is 77 km/h. Writing the distance as 10v makes the leftover leg exactly v kilometres, so the delay equation reduces to v/(v − 7) = 11/10.
The result is f(x) = |x − 3| + 9 and the vertex moves from (1, 5) to (3, 9). Both constants change: 1 becomes 3 inside the bars and 5 becomes 9 outside them.
The result is f(x) = |x − 6| + 3, with the vertex moving from (4, −3) to (6, 3). Horizontal shifts go inside the bars and subtract; vertical shifts go outside and add.
The result is f(x) = |x − 6| + 6, moving the vertex from (4, 0) to (6, 6). With no constant term to start, the up-shift supplies the outside constant directly.
Replacing x with x + 4 gives |(x + 4) - 5| = |x - 1|, moving the vertex from x = 5 to x = 1. A left shift uses a plus sign inside the function.
A vertical shift subtracts outside the absolute value, giving |x + 2| - 3. The vertex drops from (-2, 0) to (-2, -3), and the graph now crosses the x-axis twice.
The result is f(x) = |x + 8| + 8. A horizontal shift goes inside the absolute value with the opposite sign; a vertical shift goes outside with the same sign.
The result is f(x) = |x − 2| + 4 with vertex (2, 4). Starting from the parent |x| makes the rule visible: h goes inside with a minus sign, k goes outside with a plus.
The speeds are 12 km/h and 15 km/h. Setting 120/x − 120/(x+3) = 2 clears to x² + 3x − 180 = 0, whose positive root is x = 12.
Answer: x = 3/5 and y = 9/5. The second equation gives x = y/3 with no fractions to clear, and substituting turns the first into (10/3)y = 6 in one step.
There is no solution. Both expressions have slope 0.055, so their difference is the constant 2440 for every x — the graphs are parallel lines that never meet.
The answer is 4 days. Two days alone leaves 3/5 of the house, and the combined rate of 3/10 per day clears that remainder in exactly 2 more days.
The line is y = −3x + 689. The equation is two-point form for the points (362, −397) and (366, −409); cross-multiplying and dividing by 4 gives slope −3.
Every real m works. Rearranging gives m ≤ x² − x + 1, and on x > 0 that expression is unbounded above, so a large enough positive x always satisfies the inequality.
The balance after n weeks is 42 + 8(n - 1) = 8n + 34. Solving 8n + 34 >= 400 gives n >= 45.75, so week 46 with 402 dollars is the first that suffices.
The condition says z^2025 equals its own conjugate, so z^2025 is real. In polar form it forces arg(z) = k*pi/2025 for an integer k, or z = 0, with the modulus free.
If n is odd, a + b is always a factor with alternating signs in the cofactor. If n is even there is no general factorisation, though n = 6 works via cubes.
A step function. Because each additional ounce or any part of one costs a full 0.20, the total jumps at whole-ounce boundaries and stays flat in between.
From u_5 - u_2 = 3d we get d = 6 and u_1 = -5. Solving -5 + 6(n - 1) = 103 gives n = 19, and u_19 = -5 + 108 = 103 confirms it.
The discriminant is -16, which is negative, so there are no real roots and no real linear factors. Over the complex numbers it splits as 5(x - r)(x - r-bar).
Yes: subtracting uₙ₋₁ = u₁ + (n−2)d from uₙ = u₁ + (n−1)d leaves exactly d. The identity characterises arithmetic sequences and fails for any other kind.
Neither 1 nor -1 is a root, so it is irreducible over the rationals. Its derivative 3x^2 - 6x + 5 is always positive, so the cubic is strictly increasing.
All four candidates from the Rational Root Theorem fail, so it is irreducible over the rationals - yet it still has three real roots, near -0.391, 1.227 and 4.164.
x = 3 is excluded because it makes the denominator (x−3)(x+3) zero, giving the undefined form 0/0. Cancelling x − 3 leaves a hole at x = 3, not a valid value.
It does not factor. Read as a quadratic in x, its discriminant is 16y(y - 1), which is not a perfect square polynomial, unlike the familiar x^2 - 4xy + 4y^2.
256/625 = (4/5)^4, because 256 = 4^4 and 625 = 5^4. Since both parts share the exponent 4, the fraction folds into one power of 4/5, or 2^8/5^4.
The cubic equals (3x − 2)³, so the only root is x = 2/3, repeated three times. The coefficients match the binomial pattern a³ − 3a²b + 3ab² − b³ exactly.
Answer: 81/16 = (3/2)^4. Factor 81 = 3^4 and 16 = 2^4, then use a^n/b^n = (a/b)^n. As a decimal 81/16 = 5.0625, which is no integer power of a whole number.
The answer is x + 66 when x >= -66, and -x - 66 when x < -66. The split point is where the inside changes sign, found by solving x + 66 = 0 for x.
The answer is x^2 for x >= 0 and -x^2 for x < 0. Splitting at zero removes the absolute value, and the result is an odd, everywhere-increasing function.
The equation is 2((2w + 2) + w) = 64, giving width 10 m and length 22 m. The key is turning 'two more than twice the width' into 2w + 2, not 2(w + 2).
The equation is 2((3w + 4) + w) = 96, which solves to width 11 m and length 37 m. Same translation pattern as the 2w + 2 version but with a different multiplier.