Equation Solver

An equation is a mathematical statement asserting that two expressions are equal, connected by the $=$ sign:

$\text{left side} = \text{right side}$

Solving an equation means finding all values of the variable(s) that make the statement true. These values are called solutions or roots.

Equations come in many types:

• Linear: $3x + 2 = 11$
• Quadratic: $x^2 - 4x + 3 = 0$
• Rational: $\frac{x+1}{x-2} = 3$
• Radical: $\sqrt{2x+1} = x - 1$
• Exponential: $2^x = 32$
• Logarithmic: $\log_2(x) = 5$
• Absolute value: $|3x - 2| = 7$
• Trigonometric: $\sin(x) = \frac{1}{2}$

This general-purpose solver handles all these types and more, choosing the appropriate method based on the equation's structure. Unlike specialized solvers (for just linear or just quadratic), this tool identifies the equation type and applies the best strategy automatically.

1. Rational Equations

Multiply both sides by the LCD, solve the resulting polynomial, then check for extraneous solutions (values that make a denominator zero).

Example: $\frac{x+1}{x-2} = 3$

1. Multiply both sides by $(x-2)$: $x + 1 = 3(x-2)$
2. $x + 1 = 3x - 6$ → $-2x = -7$ → $x = \frac{7}{2}$
3. Check: $x = \frac{7}{2} \neq 2$ ✓

2. Radical Equations

Isolate the radical, then square (or raise to appropriate power) both sides. Always verify solutions.

Example: $\sqrt{2x+1} = x - 1$

1. Square both sides: $2x + 1 = (x-1)^2 = x^2 - 2x + 1$
2. Rearrange: $x^2 - 4x = 0$ → $x(x-4) = 0$ → $x = 0$ or $x = 4$
3. Check $x = 0$: $\sqrt{1} = -1$? No! Extraneous.
4. Check $x = 4$: $\sqrt{9} = 3$ ✓

3. Exponential Equations

If bases can be matched, equate exponents. Otherwise, take logarithms.

Example: $2^x = 32 = 2^5$ → $x = 5$

4. Absolute Value Equations

Split into two cases: the expression inside equals $+c$ or $-c$.

Example: $|3x - 2| = 7$

• Case 1: $3x - 2 = 7$ → $x = 3$
• Case 2: $3x - 2 = -7$ → $x = -\frac{5}{3}$

5. Logarithmic Equations

Convert to exponential form or use log properties to combine.

Example: $\log_2(x) = 5$ → $x = 2^5 = 32$

• Not checking for extraneous solutions: Squaring both sides or multiplying by variable expressions can introduce false solutions. Always substitute back into the original equation.
• Forgetting domain restrictions: Logarithms require positive arguments; square roots require non-negative radicands; fractions require non-zero denominators.
• Dropping solutions with absolute value: $|x| = 5$ has TWO solutions ($x = 5$ and $x = -5$). Do not forget the negative case.
• Incorrect log/exponential manipulation: $\log(a+b) \neq \log(a) + \log(b)$. The log of a sum is NOT the sum of logs.
• Dividing by a variable without checking if it's zero: If you divide both sides by $x$, you might lose the solution $x = 0$.