Question

complete the square to indentify the vertex of y=x^2 - 16x + 14

Answer 1

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

1. **Identify the coefficients**: In the expression \( x^2 - 16x + 14 \), the coefficient of \( x \) is \(-16\).

2. **Take half of the coefficient of \( x \)**: Half of \(-16\) is \(-8\).

3. **Square this value**: Squaring \(-8\) gives \( (-8)^2 = 64 \).

4. **Rewrite the quadratic**: We can add and subtract 64 in the expression to complete the square. Thus, we rewrite the equation as follows:

\[
y = (x^2 - 16x + 64) - 64 + 14
\]

5. **Factor the perfect square**: The expression \( x^2 - 16x + 64 \) can be factored as \( (x - 8)^2 \):

\[
y = (x - 8)^2 - 64 + 14
\]

6. **Combine the constants**: Simplifying the constants:

\[
-64 + 14 = -50
\]

So, we have:

\[
y = (x - 8)^2 - 50
\]

From this form, we can identify the vertex of the parabola. The vertex \( (h, k) \) is given by the point \( (8, -50) \).

Thus, the vertex of the quadratic \( y = x^2 - 16x + 14 \) is:

\[
\boxed{(8, -50)}
\]