Question

Complete the square to identify the vertex of y = x^2 - 16x + 14
A. (8,30)
B. (8,14)
C. (8,-50)
D. (8.78)

Answer 1

To complete the square for the quadratic equation \( y = x^2 - 16x + 14 \), we proceed as follows:

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

2. We need to complete the square for the expression \( x^2 - 16x \). To do this, take half of the coefficient of \( x \) (which is \(-16\)), square it, and add/subtract that term inside the equation.
- Half of \(-16\) is \(-8\).
- Squaring it gives \((-8)^2 = 64\).

3. Rewrite the equation by adding and subtracting \( 64 \):
\[
y = (x^2 - 16x + 64) - 64 + 14
\]

4. This simplifies to:
\[
y = (x - 8)^2 - 50
\]

5. Now, the equation is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( h = 8 \) and \( k = -50 \).

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

The answer is:
**C. (8, -50)**.