Question

completing the square
write p(x) = 21 + 24x + 6x² in vertex form.
p(x) = (x + )² −

Explanation:

Step1: Rewrite the quadratic function

First, we rewrite the given quadratic function \( p(x) = 6x^{2}+24x + 21\) in the standard form \(ax^{2}+bx + c\). Here, \(a = 6\), \(b = 24\), and \(c = 21\). We factor out the coefficient of \(x^{2}\) from the first two terms:
\(p(x)=6(x^{2}+4x)+21\)

Step2: Complete the square inside the parentheses

To complete the square for the expression \(x^{2}+4x\), we take half of the coefficient of \(x\) (which is \(4\)), square it, and add and subtract it inside the parentheses. Half of \(4\) is \(2\), and squaring \(2\) gives \(4\). So we have:
\(p(x)=6(x^{2}+4x + 4-4)+21\)
We can rewrite \(x^{2}+4x + 4\) as \((x + 2)^{2}\), so the expression becomes:
\(p(x)=6((x + 2)^{2}-4)+21\)

Step3: Distribute and simplify

Now we distribute the \(6\) across the terms inside the parentheses:
\(p(x)=6(x + 2)^{2}-6\times4+21\)
Calculating \(6\times4 = 24\), we get:
\(p(x)=6(x + 2)^{2}-24 + 21\)
Simplifying \(-24 + 21=-3\), we have:
\(p(x)=6(x + 2)^{2}-3\)

Answer:

\(p(x)=\boldsymbol{6}(x+\boldsymbol{2})^{2}-\boldsymbol{3}\)