Question

Complete the square to identify the vertex of y = x^2 − 16x + 14.
A) (8, 78)
B) (8, 30)
C) (8, 14)
D) (8, −50)

Answer 1

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

1. **Identify the coefficient of \( x \)**: Here, the coefficient of \( x \) is \(-16\).

2. **Take half of that coefficient and square it**:
\[
(-16/2)^2 = (-8)^2 = 64
\]

3. **Add and subtract this square inside the equation**:
\[
y = (x^2 - 16x + 64) - 64 + 14
\]

4. **Rewrite the equation**: The expression \( x^2 - 16x + 64 \) can be factored as \( (x - 8)^2 \).
\[
y = (x - 8)^2 - 64 + 14
\]
\[
y = (x - 8)^2 - 50
\]

5. **Identify the vertex**: The vertex form of a quadratic equation \( y = a(x - h)^2 + k \) shows that the vertex is at the point \( (h, k) \). In our equation:
- \( h = 8 \)
- \( k = -50 \)

Thus, the vertex of the parabola described by the equation \( y = x^2 - 16x + 14 \) is at the point \( (8, -50) \).

Therefore, the correct answer is:
**D) (8, −50)**.