Question

complete the square to re - write the quadratic function in vertex form:
$y = 9x^2 - 36x + 39$

Explanation:

Step1: Factor out the coefficient of \(x^2\) from the first two terms

First, we factor out the coefficient of \(x^2\) (which is 9) from the terms involving \(x\). So we have:
\(y = 9(x^2 - 4x) + 39\)

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 add and subtract 4 inside the parentheses:
\(y = 9((x^2 - 4x + 4) - 4) + 39\)

Step3: Simplify the expression

Now we simplify the expression. First, we recognize that \(x^2 - 4x + 4\) is a perfect square trinomial, which factors to \((x - 2)^2\). Then we distribute the 9:
\(y = 9((x - 2)^2 - 4) + 39\)
\(y = 9(x - 2)^2 - 36 + 39\)

Step4: Combine the constant terms

Finally, we combine the constant terms -36 and 39:
\(y = 9(x - 2)^2 + 3\)

Answer:

\(y = 9(x - 2)^2 + 3\)