Find the Vertex Form y=x^2+16x+65

$y = x^{2} + 16 x + 65$

Step 1

Complete the square for $x^{2} + 16 x + 65$ .

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

$a = 1$

$b = 16$

$c = 65$

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 $d = \frac{b}{2 a}$ .

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

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

Factor $2$ out of $16$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 8}{2 (1)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{8}{1}$

Divide $8$ by $1$ .

$d = 8$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 65 - \frac{16^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

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

$e = 65 - \frac{256}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 65 - \frac{256}{4}$

Divide $256$ by $4$ .

$e = 65 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 65 - 64$

Subtract $64$ from $65$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 8)}^{2} + 1$ .

${(x + 8)}^{2} + 1$

Step 2

Set $y$ equal to the new right side.

$y = {(x + 8)}^{2} + 1$