Certainly! Let's rewrite the given function [tex]\( f(x) = (3x - 9)(x + 1) \)[/tex] in vertex form by completing the square. Follow these steps:
1. First, expand the equation:
[tex]\[
f(x) = (3x - 9)(x + 1)
\][/tex]
[tex]\[
f(x) = 3x^2 + 3x - 9x - 9
\][/tex]
[tex]\[
f(x) = 3x^2 - 6x - 9
\][/tex]
2. Now, factor out the 3 from the quadratic expression:
[tex]\[
f(x) = 3(x^2 - 2x - 3)
\][/tex]
3. To complete the square, add and subtract the square of half the coefficient of [tex]\( x \)[/tex] inside the parenthesis. The coefficient of [tex]\( x \)[/tex] is -2, so half of it is -1 and its square is 1:
[tex]\[
f(x) = 3(x^2 - 2x - 3)
\][/tex]
[tex]\[
f(x) = 3((x^2 - 2x + 1) - 1 - 3)
\][/tex]
[tex]\[
f(x) = 3((x - 1)^2 - 4)
\][/tex]
[tex]\[
f(x) = 3(x - 1)^2 - 12
\][/tex]
Therefore, the equation in vertex form is:
[tex]\[
f(x) = 3(x - 1)^2 - 12
\][/tex]
So, your final answer is:
[tex]\[
f(x) = 3(x - 1)^2 - 12
\][/tex]
