Question

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

Answer 1

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

1. **Identify the necessary components**: Start with the \( x^2 \) and \( -16x \) part.

\[
y = x^2 - 16x + 14
\]

2. **Complete the square for \( x^2 - 16x \)**:
- Take the coefficient of \( x \), which is \(-16\), divide it by 2, and then square it:
\[
\left(-\frac{16}{2}\right)^2 = (-8)^2 = 64
\]

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

4. **Simplify**:
- 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 equation is now in the 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 represented by the equation \( y = x^2 - 16x + 14 \) is \((8, -50)\).