Question

complete the square to re-write the quadratic function in vertex form:
$y = -5x^2 + 30x - 43$

Explanation:

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

First, we factor out \(-5\) from the terms involving \(x\). So we have:
\(y = -5(x^2 - 6x) - 43\)

Step2: Complete the square inside the parentheses

To complete the square for \(x^2 - 6x\), we take half of the coefficient of \(x\) (which is \(-6\)), square it. Half of \(-6\) is \(-3\), and squaring \(-3\) gives \(9\). So we add and subtract \(9\) inside the parentheses (but we have to be careful with the factored out \(-5\)):
\(y = -5(x^2 - 6x + 9 - 9) - 43\)
We can rewrite the expression inside the parentheses as a perfect square minus \(9\):
\(y = -5((x - 3)^2 - 9) - 43\)

Step3: Distribute the \(-5\) and simplify

Now we distribute the \(-5\) across the terms inside the parentheses:
\(y = -5(x - 3)^2 + 45 - 43\)
Then we simplify the constant terms:
\(y = -5(x - 3)^2 + 2\)

Answer:

\(y = -5(x - 3)^2 + 2\)