Question

solve the equation using a table.
$x^2 - 12x + 32 = 0$

$x = \square$
(use a comma to separate answers as needed.)

Explanation:

Step1: Define the quadratic function

Let $f(x) = x^2 - 12x + 32$

Step2: Identify vertex x-value

The x-coordinate of the vertex of $f(x)=ax^2+bx+c$ is $x=-\frac{b}{2a}$. Here $a=1$, $b=-12$, so:
$x = -\frac{-12}{2\times1} = 6$

Step3: Create table around vertex

Calculate $f(x)$ for values near $x=6$:

• $x=4$: $f(4)=4^2 -12(4)+32=16-48+32=0$
• $x=5$: $f(5)=5^2 -12(5)+32=25-60+32=-3$
• $x=6$: $f(6)=6^2 -12(6)+32=36-72+32=-4$
• $x=7$: $f(7)=7^2 -12(7)+32=49-84+32=-3$
• $x=8$: $f(8)=8^2 -12(8)+32=64-96+32=0$

Step4: Find x where f(x)=0

Locate x-values where $f(x)=0$ from the table.

Answer:

4, 8