Complete the square to identify the vertex of y=x^2−16x+14 .(1 point)

Question

Responses

(8,78)
left parenthesis 8 comma 78 right parenthesis

(8,14)
left parenthesis 8 comma 14 right parenthesis

(8,−50)
left parenthesis 8 comma negative 50 right parenthesis

(8,30)

Answer 1

To complete the square for the quadratic equation \( y = x^2 - 16x + 14 \), we 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 and subtract this value inside the equation. Half of \(-16\) is \(-8\), and squaring it gives \(64\).

3. Rewrite the equation: \[ y = (x^2 - 16x + 64) - 64 + 14 \] This simplifies to: \[ y = (x - 8)^2 - 50 \]

4. Now the equation is in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From our equation, we see that:

   • \( h = 8 \)
   • \( k = -50 \)

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

The correct response is: (8,−50).