Question

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

Explanation:

Step1: Group the x-terms

We start with the quadratic function \( y = x^2 + 2x + 8 \). Group the terms with \( x \): \( y=(x^2 + 2x)+8 \)

Step2: Complete the square for the x-terms

To complete the square for \( x^2 + 2x \), we take half of the coefficient of \( x \) (which is \( 2 \)), so \( \frac{2}{2}=1 \), and then square it: \( 1^2 = 1 \). We add and subtract this inside the parentheses: \( y=(x^2 + 2x + 1 - 1)+8 \)

Step3: Rewrite as a perfect square and simplify

Rewrite \( x^2 + 2x + 1 \) as \( (x + 1)^2 \): \( y=((x + 1)^2 - 1)+8 \). Then simplify the constants: \( y=(x + 1)^2 - 1 + 8=(x + 1)^2 + 7 \)

Answer:

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