To determine the [tex]\(x\)[/tex]-intercept of the function [tex]\(f(x) = x^2 - 25\)[/tex], we need to find the value of [tex]\(x\)[/tex] for which [tex]\(f(x) = 0\)[/tex]. This is the point where the graph of the function intersects the [tex]\(x\)[/tex]-axis.
1. Start by setting the function equal to zero:
[tex]\[
x^2 - 25 = 0
\][/tex]
2. Solve the equation for [tex]\(x\)[/tex].
Notice that this is a difference of squares, which can be factored as follows:
[tex]\[
x^2 - 25 = (x - 5)(x + 5) = 0
\][/tex]
3. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 5 = 0 \quad \text{or} \quad x + 5 = 0
\][/tex]
[tex]\[
x = 5 \quad \text{or} \quad x = -5
\][/tex]
So, the solutions to the equation [tex]\(x^2 - 25 = 0\)[/tex] are [tex]\(x = 5\)[/tex] and [tex]\(x = -5\)[/tex].
Now, let's consider the provided choices:
A. -25
B. -15
C. -5
D. -20
Among the given choices, the value [tex]\(x = -5\)[/tex] is one of the solutions. Therefore, the [tex]\(x\)[/tex]-intercept of the function [tex]\(f(x) = x^2 - 25\)[/tex] that matches the given options is:
C. -5
