To find the [tex]\(x\)[/tex]-intercepts of the quadratic function [tex]\(f(x) = (x + 6)(x - 3)\)[/tex], we need to determine the values of [tex]\(x\)[/tex] for which [tex]\(f(x) = 0\)[/tex].
Here are the detailed steps:
1. Set the function equal to zero:
[tex]\[ (x + 6)(x - 3) = 0 \][/tex]
2. Apply the zero-product property, which states that if a product of two factors is zero, then at least one of the factors must be zero:
[tex]\[ x + 6 = 0 \quad \text{or} \quad x - 3 = 0 \][/tex]
3. Solve each equation for [tex]\(x\)[/tex]:
[tex]\[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \][/tex]
[tex]\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \][/tex]
These solutions give us the [tex]\(x\)[/tex]-values of the [tex]\(x\)[/tex]-intercepts. The [tex]\(x\)[/tex]-intercepts are the points where the graph crosses the [tex]\(x\)[/tex]-axis, and their coordinates are:
[tex]\[ (-6, 0) \quad \text{and} \quad (3, 0) \][/tex]
Among the given choices:
- [tex]\((0, -6)\)[/tex]: This point is not an [tex]\(x\)[/tex]-intercept.
- [tex]\((6, 0)\)[/tex]: This point is not an [tex]\(x\)[/tex]-intercept.
- [tex]\((-6, 0)\)[/tex]: This point is an [tex]\(x\)[/tex]-intercept.
Thus, the point [tex]\((-6, 0)\)[/tex] is an [tex]\(x\)[/tex]-intercept of the quadratic function [tex]\(f(x) = (x + 6)(x - 3)\)[/tex].
