Let's determine the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(f(x) = (x+6)(x-3)\)[/tex].
An [tex]\(x\)[/tex]-intercept occurs where the function [tex]\(f(x)\)[/tex] equals zero, that is, where:
[tex]\[ f(x) = 0 \][/tex]
Given the function:
[tex]\[ f(x) = (x+6)(x-3) \][/tex]
We set it equal to zero:
[tex]\[ (x+6)(x-3) = 0 \][/tex]
For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we solve for [tex]\(x\)[/tex] in each of the individual factors:
1. Set [tex]\(x + 6 = 0\)[/tex]:
[tex]\[ x + 6 = 0 \][/tex]
[tex]\[ x = -6 \][/tex]
2. Set [tex]\(x - 3 = 0\)[/tex]:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the solutions (the [tex]\(x\)[/tex]-intercepts) are:
- [tex]\((-6,0)\)[/tex]
- [tex]\((3,0)\)[/tex]
We now compare these with the given options:
- [tex]\((0,6)\)[/tex]
- [tex]\((0,-6)\)[/tex]
- [tex]\((6,0)\)[/tex]
- [tex]\((-6,0)\)[/tex]
The correct point that is an [tex]\(x\)[/tex]-intercept of the quadratic function [tex]\(f(x) = (x+6)(x-3)\)[/tex] is:
[tex]\[ (-6, 0) \][/tex]
