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:

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