Question

complete the square to re - write the quadratic function in vertex form: $y = 4x^{2}+8x - 3$

Explanation:

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

We have \(y = 4x^2 + 8x - 3\). Factor out 4 from the first two terms: \(y = 4(x^2 + 2x) - 3\)

Step2: Complete the square inside the parentheses

To complete the square for \(x^2 + 2x\), we take half of the coefficient of \(x\) (which is \(2\)), square it (\((\frac{2}{2})^2 = 1\)), and add and subtract it inside the parentheses. But since there is a factor of 4 outside, we need to be careful. We add \(1\) inside the parentheses and subtract \(4\times1\) outside to keep the equation balanced:
\[

$$\begin{align*} y&= 4(x^2 + 2x + 1 - 1) - 3\\ &= 4((x + 1)^2 - 1) - 3 \end{align*}$$

\]

Step3: Distribute and simplify

Distribute the 4: \(y = 4(x + 1)^2 - 4 - 3\)
Simplify the constants: \(y = 4(x + 1)^2 - 7\)

Answer:

\(y = 4(x + 1)^2 - 7\)