Find the Standard Form of the Parabola x=40y^2-240y+120

$x = 40 y^{2} - 240 y + 120$

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.

$40 y^{2} - 240 y + 120 = x$

Step 2

Subtract $120$ from both sides of the equation.

$40 y^{2} - 240 y = x - 120$

Step 3

Complete the square.

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

$a = 40$

$b = - 240$

$c = 0 = x - 120$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = x - 120$

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

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

$d = \frac{- 240}{2 \cdot 40}$

Simplify the right side.

Cancel the common factor of $- 240$ and $2$ .

Factor $2$ out of $- 240$ .

$d = \frac{2 \cdot - 120}{2 \cdot 40}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 40$ .

$d = \frac{2 \cdot - 120}{2 (40)}$

Cancel the common factor.

$d = \frac{2 \cdot - 120}{2 \cdot 40}$

Rewrite the expression.

$d = \frac{- 120}{40}$

Cancel the common factor of $- 120$ and $40$ .

Factor $40$ out of $- 120$ .

$d = \frac{40 \cdot - 3}{40}$

Cancel the common factors.

Factor $40$ out of $40$ .

$d = \frac{40 \cdot - 3}{40 (1)}$

Cancel the common factor.

$d = \frac{40 \cdot - 3}{40 \cdot 1}$

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

$e = 0 - \frac{{(- 240)}^{2}}{4 \cdot 40}$

Simplify the right side.

Simplify each term.

Raise $- 240$ to the power of $2$ .

$e = 0 - \frac{57600}{4 \cdot 40}$

Multiply $4$ by $40$ .

$e = 0 - \frac{57600}{160}$

Divide $57600$ by $160$ .

$e = 0 - 1 \cdot 360$

Multiply $- 1$ by $360$ .

$e = 0 - 360$

Subtract $360$ from $0$ .

$e = - 360$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $40 {(y - 3)}^{2} - 360$ .

$40 {(y - 3)}^{2} - 360 = x - 120$

Step 4

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

Add $360$ to both sides of the equation.

$40 {(y - 3)}^{2} = x - 120 + 360$

Add $- 120$ and $360$ .

$40 {(y - 3)}^{2} = x + 240$

Step 5

Divide each term in $40 {(y - 3)}^{2} = x + 240$ by $40$ and simplify.

Divide each term in $40 {(y - 3)}^{2} = x + 240$ by $40$ .

$\frac{40 {(y - 3)}^{2}}{40} = \frac{x}{40} + \frac{240}{40}$

Simplify the left side.

Cancel the common factor of $40$ .

Cancel the common factor.

$\frac{40 {(y - 3)}^{2}}{40} = \frac{x}{40} + \frac{240}{40}$

Divide ${(y - 3)}^{2}$ by $1$ .

${(y - 3)}^{2} = \frac{x}{40} + \frac{240}{40}$

Simplify the right side.

Divide $240$ by $40$ .

${(y - 3)}^{2} = \frac{x}{40} + 6$

Step 6

Factor.

To write $6$ as a fraction with a common denominator, multiply by $\frac{40}{40}$ .

${(y - 3)}^{2} = \frac{x}{40} + 6 \cdot \frac{40}{40}$

Combine $6$ and $\frac{40}{40}$ .

${(y - 3)}^{2} = \frac{x}{40} + \frac{6 \cdot 40}{40}$

Combine the numerators over the common denominator.

${(y - 3)}^{2} = \frac{x + 6 \cdot 40}{40}$

Multiply $6$ by $40$ .

${(y - 3)}^{2} = \frac{x + 240}{40}$

Reorder terms.

${(y - 3)}^{2} = \frac{1}{40} (x + 240)$

Find the Standard Form of the Parabola x=40y^2-240y+120

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