To determine the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(f(x) = (x-8)(x+9)\)[/tex], we follow these steps:
1. Set the function equal to zero to find the [tex]\(x\)[/tex]-intercepts:
[tex]\[
f(x) = (x-8)(x+9) = 0
\][/tex]
2. Solve for [tex]\(x\)[/tex] by setting each factor equal to zero:
[tex]\[
x - 8 = 0 \quad \text{or} \quad x + 9 = 0
\][/tex]
Solve each equation separately:
- For [tex]\(x - 8 = 0\)[/tex]:
[tex]\[
x = 8
\][/tex]
- For [tex]\(x + 9 = 0\)[/tex]:
[tex]\[
x = -9
\][/tex]
3. Determine the coordinates of the [tex]\(x\)[/tex]-intercepts:
- The point corresponding to [tex]\(x = 8\)[/tex] is [tex]\((8, 0)\)[/tex].
- The point corresponding to [tex]\(x = -9\)[/tex] is [tex]\((-9, 0)\)[/tex].
Among the given options:
- [tex]\((0, 8)\)[/tex]: Incorrect, as both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates do not match.
- [tex]\((0, -8)\)[/tex]: Incorrect, because the [tex]\(x\)[/tex]-coordinate should be either 8 or -9.
- [tex]\((9, 0)\)[/tex]: Incorrect, because [tex]\(9\)[/tex] is not a solution to the equations [tex]\(x - 8 = 0\)[/tex] or [tex]\(x + 9 = 0\)[/tex].
- [tex]\((-9, 0)\)[/tex]: Correct, since [tex]\((-9, 0)\)[/tex] is one of the [tex]\(x\)[/tex]-intercepts.
So, the correct point which is an [tex]\(x\)[/tex]-intercept of the quadratic function [tex]\(f(x) = (x-8)(x+9)\)[/tex] is [tex]\(\boxed{(-9, 0)}\)[/tex].
