Find the Standard Form of the Parabola y=2x^2+16x+18

$y = 2 x^{2} + 16 x + 18$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} + 16 x + 18 = y$

Step 2

Subtract $18$ from both sides of the equation.

$2 x^{2} + 16 x = y - 18$

Step 3

Complete the square.

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

$a = 2$

$b = 16$

$c = 0 = y - 18$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 18$

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

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

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

Simplify the right side.

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

Factor $2$ out of $16$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $8$ and $2$ .

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 4}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{4}{1}$

Divide $4$ by $1$ .

$d = 4$

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

$e = 0 - \frac{16^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{256}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{256}{8}$

Divide $256$ by $8$ .

$e = 0 - 1 \cdot 32$

Multiply $- 1$ by $32$ .

$e = 0 - 32$

Subtract $32$ from $0$ .

$e = - 32$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + 4)}^{2} - 32$ .

$2 {(x + 4)}^{2} - 32 = y - 18$

Step 4

Move all terms not containing $x$ to the right side of the equation.

Add $32$ to both sides of the equation.

$2 {(x + 4)}^{2} = y - 18 + 32$

Add $- 18$ and $32$ .

$2 {(x + 4)}^{2} = y + 14$

Step 5

Divide each term in $2 {(x + 4)}^{2} = y + 14$ by $2$ and simplify.

Divide each term in $2 {(x + 4)}^{2} = y + 14$ by $2$ .

$\frac{2 {(x + 4)}^{2}}{2} = \frac{y}{2} + \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 {(x + 4)}^{2}}{2} = \frac{y}{2} + \frac{14}{2}$

Divide ${(x + 4)}^{2}$ by $1$ .

${(x + 4)}^{2} = \frac{y}{2} + \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

${(x + 4)}^{2} = \frac{y}{2} + 7$

Step 6

Factor.

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x + 4)}^{2} = \frac{y}{2} + 7 \cdot \frac{2}{2}$

Combine $7$ and $\frac{2}{2}$ .

${(x + 4)}^{2} = \frac{y}{2} + \frac{7 \cdot 2}{2}$

Combine the numerators over the common denominator.

${(x + 4)}^{2} = \frac{y + 7 \cdot 2}{2}$

Multiply $7$ by $2$ .

${(x + 4)}^{2} = \frac{y + 14}{2}$

Reorder terms.

${(x + 4)}^{2} = \frac{1}{2} (y + 14)$

Find the Standard Form of the Parabola y=2x^2+16x+18

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