Find the Standard Form of the Parabola x=2y^2-4y-6

$x = 2 y^{2} - 4 y - 6$

Step 1

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

$2 y^{2} - 4 y - 6 = x$

Step 2

Add $6$ to both sides of the equation.

$2 y^{2} - 4 y = x + 6$

Step 3

Complete the square.

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

$a = 2$

$b = - 4$

$c = 0 = x + 6$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = x + 6$

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

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

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

Simplify the right side.

Cancel the common factor of $- 4$ and $2$ .

Factor $2$ out of $- 4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$d = - 1$

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot 2}$

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 2}$

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot 2}$

Multiply $4^{2}$ by $1$ .

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

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

$e = 0 - \frac{2 \cdot 2}{2 (1)}$

Cancel the common factor.

$e = 0 - \frac{2 \cdot 2}{2 \cdot 1}$

Rewrite the expression.

$e = 0 - \frac{2}{1}$

Divide $2$ by $1$ .

$e = 0 - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$e = 0 - 2$

Subtract $2$ from $0$ .

$e = - 2$

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

$2 {(y - 1)}^{2} - 2 = x + 6$

Step 4

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

Add $2$ to both sides of the equation.

$2 {(y - 1)}^{2} = x + 6 + 2$

Add $6$ and $2$ .

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

Step 5

Divide each term in $2 {(y - 1)}^{2} = x + 8$ by $2$ and simplify.

Divide each term in $2 {(y - 1)}^{2} = x + 8$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide ${(y - 1)}^{2}$ by $1$ .

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

Simplify the right side.

Divide $8$ by $2$ .

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

Step 6

Factor.

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine $4$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Multiply $4$ by $2$ .

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

Reorder terms.

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

Find the Standard Form of the Parabola x=2y^2-4y-6

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