Find the Vertex Form y=-x^2+16x-66

$y = - x^{2} + 16 x - 66$

Step 1

Complete the square for $- x^{2} + 16 x - 66$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 16$

$c = - 66$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{16}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $16$ and $2$ .

Factor $2$ out of $16$ .

$d = \frac{2 \cdot 8}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{8}{- 1}$ .

$d = - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$d = - 8$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 66 - \frac{16^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $16$ to the power of $2$ .

$e = - 66 - \frac{256}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 66 - \frac{256}{- 4}$

Divide $256$ by $- 4$ .

$e = - 66 - - 64$

Multiply $- 1$ by $- 64$ .

$e = - 66 + 64$

Add $- 66$ and $64$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 8)}^{2} - 2$ .

$- {(x - 8)}^{2} - 2$

Step 2

Set $y$ equal to the new right side.

$y = - {(x - 8)}^{2} - 2$