To rewrite the equation [tex]\( f(x) = (3x - 9)(x + 1) \)[/tex] in vertex form by completing the square, follow these steps:
1. Expand the equation:
[tex]\[
f(x) = (3x - 9)(x + 1)
\][/tex]
Distribute the terms:
[tex]\[
f(x) = 3x(x + 1) - 9(x + 1)
\][/tex]
[tex]\[
f(x) = 3x^2 + 3x - 9x - 9
\][/tex]
Combine like terms:
[tex]\[
f(x) = 3x^2 - 6x - 9
\][/tex]
2. Rewrite the quadratic in the form [tex]\( ax^2 + bx + c \)[/tex]:
[tex]\[
f(x) = 3x^2 - 6x - 9
\][/tex]
3. Complete the square:
- Factor out the coefficient of [tex]\( x^2 \)[/tex] (which is 3) from the first two terms:
[tex]\[
f(x) = 3(x^2 - 2x) - 9
\][/tex]
- To complete the square inside the parentheses, take half the coefficient of [tex]\( x \)[/tex] (which is [tex]\(-2\)[/tex]), square it, and add and subtract this square inside the parentheses. Half of [tex]\(-2\)[/tex] is [tex]\(-1\)[/tex], and [tex]\((-1)^2\)[/tex] is 1:
[tex]\[
f(x) = 3(x^2 - 2x + 1 - 1) - 9
\][/tex]
- Simplify the terms inside the parentheses:
[tex]\[
f(x) = 3((x - 1)^2 - 1) - 9
\][/tex]
- Distribute the 3:
[tex]\[
f(x) = 3(x - 1)^2 - 3 - 9
\][/tex]
- Combine constants:
[tex]\[
f(x) = 3(x - 1)^2 - 12
\][/tex]
So, the equation in vertex form is:
[tex]\[
f(x) = 3(x - 1)^2 - 12
\][/tex]
