Find the Standard Form of the Parabola x=y^2+20y+87

$x = y^{2} + 20 y + 87$

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.

$y^{2} + 20 y + 87 = x$

Step 2

Subtract $87$ from both sides of the equation.

$y^{2} + 20 y = x - 87$

Step 3

Complete the square.

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

$a = 1$

$b = 20$

$c = 0 = x - 87$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = x - 87$

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

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

$d = \frac{20}{2 \cdot 1}$

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

$d = \frac{2 \cdot 10}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot 10}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{10}{1}$

Divide $10$ by $1$ .

$d = 10$

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

$e = 0 - \frac{20^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $20$ to the power of $2$ .

$e = 0 - \frac{400}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{400}{4}$

Divide $400$ by $4$ .

$e = 0 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Subtract $100$ from $0$ .

$e = - 100$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 10)}^{2} - 100$ .

${(y + 10)}^{2} - 100 = x - 87$

Step 4

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

Add $100$ to both sides of the equation.

${(y + 10)}^{2} = x - 87 + 100$

Add $- 87$ and $100$ .

${(y + 10)}^{2} = x + 13$

Find the Standard Form of the Parabola x=y^2+20y+87

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