Question

complete the square to re - write the quadratic function in vertex form. \\(y = - 5x^{2}+30x - 40\\)

Explanation:

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

Given the quadratic function \(y = -5x^2 + 30x - 40\), we first factor out \(-5\) from the terms involving \(x\):
\(y=-5(x^2 - 6x)-40\)

Step2: Complete the square inside the parentheses

To complete the square for the expression \(x^2 - 6x\), we take half of the coefficient of \(x\) (which is \(-6\)), square it, and add and subtract it inside the parentheses. Half of \(-6\) is \(-3\), and squaring it gives \(9\). So we have:
\(y=-5(x^2 - 6x + 9 - 9)-40\)
We can rewrite the expression inside the parentheses as a perfect square and a constant:
\(y=-5((x - 3)^2 - 9)-40\)

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

Now we distribute the \(-5\) across the terms inside the parentheses:
\(y=-5(x - 3)^2+45 - 40\)
Then we simplify the constant terms:
\(y=-5(x - 3)^2 + 5\)

Answer:

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