Question

complete the square to indentify the vertex of
y = x^2 - 16x + 14
a (8,-50)
b (8,78)
c (8,14)
d (8,30)

Answer 1

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

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

2. To complete the square, focus on the \( x^2 - 16x \) part. Take half of the coefficient of \( x \) (which is -16), square it, and add it inside the equation:
\[
\text{Half of } -16 = -8
\]
\[
\text{Square it: } (-8)^2 = 64
\]

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

4. Rewrite the equation by factoring the perfect square trinomial and simplifying:
\[
y = (x - 8)^2 - 50
\]

5. Now, we can identify the vertex of the parabola. The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From our equation:
\[
(h, k) = (8, -50)
\]

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

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