To determine the correct [tex]$x$[/tex]-intercepts of the quadratic function when its factors are given as [tex]$(x + 2)$[/tex] and [tex]$(x - 9)$[/tex], follow these steps:
1. Identify the factors:
The quadratic function can be factored into:
[tex]\[
(x + 2)(x - 9)
\][/tex]
2. Set each factor equal to zero to find the [tex]$x$[/tex]-intercepts:
The [tex]$x$[/tex]-intercepts occur where the function equals zero. This means you need to solve for [tex]$x$[/tex] in each equation formed by setting each factor to zero.
- For the factor [tex]$(x + 2) = 0$[/tex]:
[tex]\[
x + 2 = 0
\][/tex]
Solving for [tex]$x$[/tex], you get:
[tex]\[
x = -2
\][/tex]
Thus, one [tex]$x$[/tex]-intercept is [tex]$(-2, 0)$[/tex].
- For the factor [tex]$(x - 9) = 0$[/tex]:
[tex]\[
x - 9 = 0
\][/tex]
Solving for [tex]$x$[/tex], you get:
[tex]\[
x = 9
\][/tex]
Thus, the other [tex]$x$[/tex]-intercept is [tex]$(9, 0)$[/tex].
3. Conclusion:
The [tex]$x$[/tex]-intercepts of the quadratic function are at [tex]$(-2, 0)$[/tex] and [tex]$(9, 0)$[/tex].
Therefore, the correct answer is:
[tex]\( C. (-2,0) \text{ and } (9,0) \)[/tex]
