Sure! Let's go through the process of rewriting the quadratic function [tex]\( g(x) = 4x^2 + 88x \)[/tex] in vertex form step-by-step.
To convert a quadratic function into vertex form, we need to rewrite it in the form:
[tex]\[ g(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
### Step 1: Factor out the coefficient of [tex]\( x^2 \)[/tex] from the quadratic and linear terms
Given the function:
[tex]\[ g(x) = 4x^2 + 88x \][/tex]
we factor out the coefficient [tex]\( 4 \)[/tex] from the terms involving [tex]\( x \)[/tex]:
[tex]\[ g(x) = 4(x^2 + 22x) \][/tex]
### Step 2: Complete the Square
Next, we complete the square inside the parenthesis. To do this, we take the coefficient of [tex]\( x \)[/tex], divide it by 2, and square the result. This will allow us to create a perfect square trinomial.
The coefficient of [tex]\( x \)[/tex] inside the parenthesis is 22. Half of 22 is 11, and [tex]\( 11^2 = 121 \)[/tex].
Now, we rewrite the quadratic expression inside the parenthesis as a perfect square trinomial:
[tex]\[ x^2 + 22x = (x + 11)^2 - 121 \][/tex]
So, we substitute this back into our equation:
[tex]\[ g(x) = 4\left( (x + 11)^2 - 121 \right) \][/tex]
### Step 3: Simplify
Now we distribute the 4 through the parenthesis:
[tex]\[ g(x) = 4(x + 11)^2 - 4 \cdot 121 \][/tex]
[tex]\[ g(x) = 4(x + 11)^2 - 484 \][/tex]
### Conclusion
Thus, the quadratic function in vertex form is:
[tex]\[ g(x) = 4(x + 11)^2 - 484 \][/tex]
In the format requested in your question, that is:
[tex]\[ g(x) = 4(x + 11)^2 - 484 \][/tex]
So, the value of the constant term boxed is [tex]\( -484 \)[/tex].
In summary:
[tex]\[ g(x) = 4(x + 11)^2 + (-484) \][/tex]
Therefore,:
[tex]\[ g(x) = 4(x + 11)^2 - 484 \][/tex]
