Question

complete the square to re-write the quadratic function in vertex form:
y = x² - 2x + 7

Explanation:

Step1: Group x terms

Separate the quadratic and linear terms: \( y=(x^{2}-2x)+7 \)

Step2: Complete the square

Take half of the coefficient of \( x \) (which is -2), square it: \( (\frac{-2}{2})^{2}=(-1)^{2}=1 \). Add and subtract this inside the parentheses: \( y=(x^{2}-2x + 1-1)+7 \)

Step3: Rewrite as perfect square

Rewrite \( x^{2}-2x + 1 \) as \( (x - 1)^{2} \): \( y=((x - 1)^{2}-1)+7 \)

Step4: Simplify constants

Combine the constant terms: \( y=(x - 1)^{2}-1 + 7=(x - 1)^{2}+6 \)

Answer:

\( y=(x - 1)^{2}+6 \)