Find the Standard Form of the Parabola y=2x^2+12x+15

$y = 2 x^{2} + 12 x + 15$

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.

$2 x^{2} + 12 x + 15 = y$

Step 2

Subtract $15$ from both sides of the equation.

$2 x^{2} + 12 x = y - 15$

Step 3

Complete the square.

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

$a = 2$

$b = 12$

$c = 0 = y - 15$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 15$

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

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

$d = \frac{12}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot 6}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{6}{2}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 3}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{12^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{144}{8}$

Divide $144$ by $8$ .

$e = 0 - 1 \cdot 18$

Multiply $- 1$ by $18$ .

$e = 0 - 18$

Subtract $18$ from $0$ .

$e = - 18$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + 3)}^{2} - 18$ .

$2 {(x + 3)}^{2} - 18 = y - 15$

Step 4

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

Add $18$ to both sides of the equation.

$2 {(x + 3)}^{2} = y - 15 + 18$

Add $- 15$ and $18$ .

$2 {(x + 3)}^{2} = y + 3$

Step 5

Divide each term in $2 {(x + 3)}^{2} = y + 3$ by $2$ and simplify.

Divide each term in $2 {(x + 3)}^{2} = y + 3$ by $2$ .

$\frac{2 {(x + 3)}^{2}}{2} = \frac{y}{2} + \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 {(x + 3)}^{2}}{2} = \frac{y}{2} + \frac{3}{2}$

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{y}{2} + \frac{3}{2}$

Step 6

Factor.

Combine the numerators over the common denominator.

${(x + 3)}^{2} = \frac{y + 3}{2}$

Reorder terms.

${(x + 3)}^{2} = \frac{1}{2} (y + 3)$

Find the Standard Form of the Parabola y=2x^2+12x+15

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