To determine which point is an [tex]$x$[/tex]-intercept of the quadratic function [tex]\(f(x) = (x-8)(x+9)\)[/tex], we need to find the values of [tex]\(x\)[/tex] where the function equals zero, because [tex]\(x\)[/tex]-intercepts occur where [tex]\(y = 0\)[/tex].
1. Set the function equal to zero:
[tex]\[ f(x) = (x-8)(x+9) = 0 \][/tex]
2. Solve the equation:
- For the product [tex]\((x-8)(x+9)\)[/tex] to be zero, either [tex]\( (x-8) = 0 \)[/tex] or [tex]\( (x+9) = 0 \)[/tex].
- Solving the first equation [tex]\( (x-8) = 0 \)[/tex]:
[tex]\[ x - 8 = 0 \][/tex]
[tex]\[ x = 8 \][/tex]
- Solving the second equation [tex]\( (x+9) = 0 \)[/tex]:
[tex]\[ x + 9 = 0 \][/tex]
[tex]\[ x = -9 \][/tex]
3. Identify the points:
- Each [tex]\(x\)[/tex]-value found corresponds to an [tex]\(x\)[/tex]-intercept.
- The [tex]\(x\)[/tex]-intercept for [tex]\(x = 8\)[/tex] is the point [tex]\((8, 0)\)[/tex].
- The [tex]\(x\)[/tex]-intercept for [tex]\(x = -9\)[/tex] is the point [tex]\((-9, 0)\)[/tex].
4. Check the options provided:
- [tex]\((0, 8)\)[/tex]
- [tex]\((0, -8)\)[/tex]
- [tex]\((9, 0)\)[/tex]
- [tex]\((-9, 0)\)[/tex]
5. Determine the correct [tex]\(x\)[/tex]-intercept from the options:
- The only matching point is [tex]\((-9, 0)\)[/tex], which corresponds to one of the [tex]\(x\)[/tex]-intercepts we found above.
Therefore, the point that is an [tex]\(x\)[/tex]-intercept of the quadratic function [tex]\(f(x) = (x-8)(x+9)\)[/tex] is [tex]\((-9, 0)\)[/tex].
