Question

use factoring to solve the quadratic equation. check by substitution or by using a graphing utility and identifying x-intercepts.\\(x^2 - 3x - 40 = 0\\)\\(\\)\\(\\)the solution set is \\(\\{\\}\\).\\(\\) (type an integer or a simplified fraction. use a comma to separate answers as needed. type each solution only once.)

Explanation:

Step1: Factor the quadratic trinomial

We need two numbers that multiply to $-40$ and add to $-3$. These numbers are $-8$ and $5$.
$x^2 - 3x - 40 = (x - 8)(x + 5) = 0$

Step2: Apply zero product property

Set each factor equal to 0.
$x - 8 = 0$ or $x + 5 = 0$

Step3: Solve for x in each equation

For $x - 8 = 0$: $x = 8$
For $x + 5 = 0$: $x = -5$

Step4: Verify solutions (substitute back)

For $x=8$: $8^2 - 3(8) - 40 = 64 - 24 - 40 = 0$
For $x=-5$: $(-5)^2 - 3(-5) - 40 = 25 + 15 - 40 = 0$

Answer:

$\{-5, 8\}$