Solve the quadratic equation x² + 17x - 120 = 0.

Step-by-step solution

1. 1

   Try integer factoring first, and watch it fail. Factoring needs two integers whose product is $-120$ and whose sum is $+17$. Run through the factor pairs of $120$: $(24,-5)$ sums to $19$, $(30,-4)$ sums to $26$, $(20,-6)$ sums to $14$, $(40,-3)$ sums to $37$, $(15,-8)$ sums to $7$. Nothing lands on $17$, so no integer factorization exists. A frequently repeated wrong answer here is $(x+24)(x-5)$ — but that expands to $x^2+19x-120$, not $x^2+17x-120$.

2. 2

   Confirm the failure with the discriminant. For $ax^2+bx+c$ with $a=1$, $b=17$, $c=-120$,

   $\Delta=b^2-4ac=17^2-4(1)(-120)=289+480=769$

   Since $27^2=729$ and $28^2=784$, the value $769$ sits strictly between two consecutive squares, so it is not a perfect square. That is exactly the algebraic reason the integer search above could never succeed: rational roots require a square discriminant.

3. 3

   Apply the quadratic formula. With $\Delta=769>0$ there are two distinct irrational roots:

   $x=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{-17\pm\sqrt{769}}{2}$

   This is the exact answer; $769$ is prime, hence free of square factors, so $\sqrt{769}$ cannot be simplified further.

4. 4

   Convert to decimals. $\sqrt{769}\approx 27.7308$, so

   $x_1=\frac{-17+27.7308}{2}\approx 5.3654,\qquad x_2=\frac{-17-27.7308}{2}\approx -22.3654$

   Note how far $x_2$ is from the tempting $-24$: the two are not interchangeable.

5. 5

   Check with Vieta's formulas. The sum of the roots must be $-b/a=-17$ and the product must be $c/a=-120$. Indeed $5.3654+(-22.3654)=-17$, and exactly,

   $x_1x_2=\frac{(-17)^2-769}{4}=\frac{289-769}{4}=\frac{-480}{4}=-120$

   Both conditions hold, so the roots are correct.