Question

question 8 of 10
which equation represents the vertex form of the equation $y = x^2 - 12x + 38$?

a. $y = (x - 12)^2 + 2$

b. $y = (x - 12)^2 + 38$

c. $y = (x - 6)^2 + 2$

d. $y = (x - 6)^2 + 38$

Explanation:

Step1: Recall vertex form formula

The vertex form of a quadratic equation \( y = ax^2+bx + c \) is \( y=a(x - h)^2+k \), where \((h,k)\) is the vertex. To convert \( y=x^2 - 12x + 38 \) to vertex form, we complete the square.
For the quadratic expression \( x^2-12x \), we take half of the coefficient of \( x \), square it, and add and subtract it. The coefficient of \( x \) is \(- 12\), half of it is \(\frac{-12}{2}=-6\), and squaring it gives \((-6)^2 = 36\).

Step2: Complete the square

Rewrite the equation as:
\[

$$\begin{align*} y&=x^2-12x + 36+38 - 36\\ y&=(x - 6)^2+2 \end{align*}$$

\]
(We added and subtracted 36 to complete the square for the \(x^2-12x\) part. The \(x^2-12x + 36\) is a perfect square trinomial which factors to \((x - 6)^2\), and then we simplify \(38-36 = 2\).)

Answer:

C. \( y=(x - 6)^2+2 \)