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 given equation:
[tex]\[ f(x) = (3x - 9)(x + 1) \][/tex]
Using distributive property (FOIL method):
[tex]\[ f(x) = 3x \cdot x + 3x \cdot 1 - 9 \cdot x - 9 \cdot 1 \][/tex]
[tex]\[ f(x) = 3x^2 + 3x - 9x - 9 \][/tex]
[tex]\[ f(x) = 3x^2 - 6x - 9 \][/tex]
2. Identify coefficients from the standard form [tex]\( ax^2 + bx + c \)[/tex]:
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = -9 \)[/tex].
3. Rewrite the quadratic equation in the form [tex]\( a(x - h)^2 + k \)[/tex] by completing the square.
First, factor out the coefficient of [tex]\( x^2 \)[/tex], [tex]\( a = 3 \)[/tex]:
[tex]\[ f(x) = 3(x^2 - 2x) - 9 \][/tex]
4. Complete the square inside the parentheses:
Take the coefficient of [tex]\( x \)[/tex] which is [tex]\(-2\)[/tex], halve it and square it:
[tex]\[ \left(\frac{-2}{2}\right)^2 = 1 \][/tex]
Add and subtract this square inside the parentheses:
[tex]\[ f(x) = 3(x^2 - 2x + 1 - 1) - 9 \][/tex]
[tex]\[ f(x) = 3((x^2 - 2x + 1) - 1) - 9 \][/tex]
[tex]\[ f(x) = 3((x - 1)^2 - 1) - 9 \][/tex]
5. Simplify the expression:
Distribute the [tex]\( 3 \)[/tex]:
[tex]\[ f(x) = 3(x - 1)^2 - 3 - 9 \][/tex]
[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]
