Complete the square to identify the vertex of y = x^2 - 16x + 14

Question

A: ( 8, 14)

B: ( 8, 78)

C: (8, 30)

D: ( 8, -50)

Answer 1

To complete the square for the quadratic equation \( y = x^2 - 16x + 14 \), we can follow these steps:

1. Start with the equation: \[ y = x^2 - 16x + 14 \]

2. Focus on the \( x^2 - 16x \) part:

   • To complete the square, take half of the coefficient of \( x \) (which is -16), square it, and then add and subtract that square inside the equation.
   • Half of -16 is -8, and squaring that gives \( 64 \).

3. Rewrite the equation: \[ y = (x^2 - 16x + 64) - 64 + 14 \] \[ y = (x - 8)^2 - 50 \]

Now the equation is in vertex form \( y = (x - h)^2 + k \), where \( (h, k) \) is the vertex.

4. Identify \( h \) and \( k \):
   • Here \( h = 8 \) and \( k = -50 \).

Thus, the vertex is \( (8, -50) \).

The correct answer is D: (8, -50).