To determine the x-intercepts of the quadratic function [tex]\( g(x) = x^2 - 9x + 20 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( g(x) = 0 \)[/tex].
The function [tex]\( g(x) = x^2 - 9x + 20 \)[/tex] can be rephrased as:
[tex]\[ x^2 - 9x + 20 = 0 \][/tex]
This is a quadratic equation in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To find the x-intercepts, we solve this quadratic equation.
We can use the quadratic formula to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this equation:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -9 \)[/tex]
- [tex]\( c = 20 \)[/tex]
Substitute these values into the quadratic formula:
[tex]\[ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{81 - 80}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm \sqrt{1}}{2} \][/tex]
[tex]\[ x = \frac{9 \pm 1}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{9 + 1}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ x = \frac{9 - 1}{2} = \frac{8}{2} = 4 \][/tex]
Thus, the x-intercepts of the quadratic function [tex]\( g(x) = x^2 - 9x + 20 \)[/tex] are [tex]\( x = 5 \)[/tex] and [tex]\( x = 4 \)[/tex].
This corresponds to the intercept points [tex]\((5, 0)\)[/tex] and [tex]\((4, 0)\)[/tex].
Therefore, the correct answer is:
[tex]\[
(5,0) \text{ and } (4,0)
\][/tex]
