To convert the given quadratic function [tex]\( g(x) = 4x^2 + 88x \)[/tex] into vertex form, we will follow these steps:
1. Identify the coefficients: The given quadratic function is [tex]\( g(x) = ax^2 + bx + c \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = 88 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Find the vertex (h, k): The vertex form of a quadratic function is [tex]\( g(x) = a(x - h)^2 + k \)[/tex].
- Calculate [tex]\( h \)[/tex] using [tex]\( h = -\frac{b}{2a} \)[/tex]:
[tex]\[
h = -\frac{88}{2 \cdot 4} = -\frac{88}{8} = -11
\][/tex]
- Calculate [tex]\( k \)[/tex] by substituting [tex]\( h \)[/tex] back into the original quadratic function:
First, substitute [tex]\( h = -11 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(-11) = 4(-11)^2 + 88(-11)
\][/tex]
Simplify the expression:
[tex]\[
(-11)^2 = 121
\][/tex]
[tex]\[
4 \cdot 121 = 484
\][/tex]
[tex]\[
88 \cdot (-11) = -968
\][/tex]
[tex]\[
k = 484 - 968 = -484
\][/tex]
3. Rewrite the function in vertex form: The vertex form of the function is given by [tex]\( g(x) = 4(x - (-11))^2 + (-484) \)[/tex], which simplifies to:
[tex]\[
g(x) = 4(x + 11)^2 - 484
\][/tex]
Thus, the function written in vertex form is:
[tex]\[
g(x) = 4 (x + 11)^2 - 484
\][/tex]
So, the filled squares are:
[tex]\[
g(x) = \boxed{4} (x + 11)^2 + \boxed{-484}
\][/tex]
