Question

writing a quadratic function in vertex form
write ( g(x) = 4x^2 + 88x ) in vertex form.
the function written in vertex form is ( g(x) = square (x + 11)^2 + square ).

Explanation:

Step1: Recall Vertex Form and Completion

The vertex form of a quadratic function is \(g(x)=a(x - h)^2 + k\). For \(g(x)=4x^2 + 88x\), first factor out the coefficient of \(x^2\) from the first two terms.
\(g(x)=4(x^2 + 22x)\)

Step2: Complete the Square

To complete the square inside the parentheses, take half of the coefficient of \(x\) (which is \(22\)), square it. Half of \(22\) is \(11\), and \(11^2 = 121\). So we add and subtract \(121\) inside the parentheses (but we have to account for the fact that we factored out a \(4\) earlier).
\(g(x)=4(x^2 + 22x+ 121- 121)\)
\(g(x)=4((x + 11)^2-121)\)

Step3: Distribute the 4

Distribute the \(4\) across the terms inside the parentheses.
\(g(x)=4(x + 11)^2-4\times121\)
\(g(x)=4(x + 11)^2-484\)

Answer:

The first blank is \(4\) and the second blank is \(-484\), so \(g(x)=\boldsymbol{4}(x + 11)^2+\boldsymbol{-484}\) (or \(g(x)=\boldsymbol{4}(x + 11)^2\boldsymbol{-484}\))