To determine the x-intercepts of the quadratic function [tex]\( f(x) = (x-4)(x+2) \)[/tex], we need to find the values of [tex]\( x \)[/tex] where [tex]\( f(x) = 0 \)[/tex].
We start with the given quadratic function:
[tex]\[ f(x) = (x-4)(x+2) \][/tex]
To find the x-intercepts, set the function equal to zero:
[tex]\[ (x-4)(x+2) = 0 \][/tex]
This equation tells us that the product of [tex]\((x-4)\)[/tex] and [tex]\((x+2)\)[/tex] equals zero. For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
1. Solve for [tex]\( x \)[/tex] when [tex]\( x-4 = 0 \)[/tex]:
[tex]\[ x-4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
2. Solve for [tex]\( x \)[/tex] when [tex]\( x+2 = 0 \)[/tex]:
[tex]\[ x+2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
Thus, the x-intercepts of the function [tex]\( f(x) = (x-4)(x+2) \)[/tex] are at [tex]\( x = 4 \)[/tex] and [tex]\( x = -2 \)[/tex]. Each of these corresponds to a point on the graph where [tex]\( y = 0 \)[/tex].
The x-intercepts are:
[tex]\[ (4, 0) \][/tex]
[tex]\[ (-2, 0) \][/tex]
Now, let's identify which of the given points is an x-intercept from the list:
- [tex]\((-4,0)\)[/tex]
- [tex]\((-2,0)\)[/tex]
- [tex]\((0,2)\)[/tex]
- [tex]\((4,-2)\)[/tex]
From the x-intercepts we found, we see that [tex]\((4, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex] are the points where the graph intersects the x-axis.
Given the options, the point [tex]\((-2, 0)\)[/tex] matches one of the x-intercepts we found. Therefore, the correct answer is:
[tex]\((-2, 0)\)[/tex].
