To find the [tex]\(x\)[/tex]-intercepts of the function [tex]\( g(x) = x^2 - 9x + 20 \)[/tex], we need to determine the points where the graph of the function crosses the [tex]\(x\)[/tex]-axis. The [tex]\(x\)[/tex]-intercepts occur where [tex]\( g(x) = 0 \)[/tex].
Step-by-step solution:
1. Set the function [tex]\( g(x) \)[/tex] equal to zero:
[tex]\[
x^2 - 9x + 20 = 0
\][/tex]
2. Solve this quadratic equation:
To solve the quadratic equation, we look for factors of 20 that add up to -9.
3. Factor the quadratic expression:
[tex]\[
(x - 4)(x - 5) = 0
\][/tex]
4. Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x - 4 = 0 \quad \text{or} \quad x - 5 = 0
\][/tex]
Solving these equations:
[tex]\[
x = 4 \quad \text{or} \quad x = 5
\][/tex]
5. Determine the [tex]\(x\)[/tex]-intercepts:
The values of [tex]\( x \)[/tex] found indicate the points where the function crosses the [tex]\( x \)[/tex]-axis. Therefore, the [tex]\( x \)[/tex]-intercepts are at:
[tex]\[
(4, 0) \quad \text{and} \quad (5, 0)
\][/tex]
So, the [tex]\( x \)[/tex]-intercepts of [tex]\( g(x) = x^2 - 9x + 20 \)[/tex] are [tex]\((4, 0)\)[/tex] and [tex]\((5, 0)\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{(5, 0)\text{ and }(4, 0)}
\][/tex]
