Precalculus Examples

$x = 20 y^{2} - 120 y + 60$

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.

$20 y^{2} - 120 y + 60 = x$

Step 2

Subtract $60$ from both sides of the equation.

$20 y^{2} - 120 y = x - 60$

Step 3

Complete the square.

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Step 3.1

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

$a = 20$

$b = - 120$

$c = 0 = x - 60$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = x - 60$

Step 3.3

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

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Step 3.3.1

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

$d = \frac{- 120}{2 \cdot 20}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $- 120$ and $2$ .

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Step 3.3.2.1.1

Factor $2$ out of $- 120$ .

$d = \frac{2 \cdot - 60}{2 \cdot 20}$

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

Factor $2$ out of $2 \cdot 20$ .

$d = \frac{2 \cdot - 60}{2 (20)}$

Step 3.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 60}{2 \cdot 20}$

Step 3.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 60}{20}$

Step 3.3.2.2

Cancel the common factor of $- 60$ and $20$ .

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Step 3.3.2.2.1

Factor $20$ out of $- 60$ .

$d = \frac{20 \cdot - 3}{20}$

Step 3.3.2.2.2

Cancel the common factors.

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Step 3.3.2.2.2.1

Factor $20$ out of $20$ .

$d = \frac{20 \cdot - 3}{20 (1)}$

Step 3.3.2.2.2.2

Cancel the common factor.

$d = \frac{20 \cdot - 3}{20 \cdot 1}$

Step 3.3.2.2.2.3

Rewrite the expression.

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

Step 3.3.2.2.2.4

Divide $- 3$ by $1$ .

$d = - 3$

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 120)}^{2}}{4 \cdot 20}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise $- 120$ to the power of $2$ .

$e = 0 - \frac{14400}{4 \cdot 20}$

Step 3.4.2.1.2

Multiply $4$ by $20$ .

$e = 0 - \frac{14400}{80}$

Step 3.4.2.1.3

Divide $14400$ by $80$ .

$e = 0 - 1 \cdot 180$

Step 3.4.2.1.4

Multiply $- 1$ by $180$ .

$e = 0 - 180$

Step 3.4.2.2

Subtract $180$ from $0$ .

$e = - 180$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $20 {(y - 3)}^{2} - 180$ .

$20 {(y - 3)}^{2} - 180 = x - 60$

Step 4

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

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Step 4.1

Add $180$ to both sides of the equation.

$20 {(y - 3)}^{2} = x - 60 + 180$

Step 4.2

Add $- 60$ and $180$ .

$20 {(y - 3)}^{2} = x + 120$

Step 5

Divide each term in $20 {(y - 3)}^{2} = x + 120$ by $20$ and simplify.

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Step 5.1

Divide each term in $20 {(y - 3)}^{2} = x + 120$ by $20$ .

$\frac{20 {(y - 3)}^{2}}{20} = \frac{x}{20} + \frac{120}{20}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $20$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{20 {(y - 3)}^{2}}{20} = \frac{x}{20} + \frac{120}{20}$

Step 5.2.1.2

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

${(y - 3)}^{2} = \frac{x}{20} + \frac{120}{20}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $120$ by $20$ .

${(y - 3)}^{2} = \frac{x}{20} + 6$

Step 6

Factor.

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Step 6.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

${(y - 3)}^{2} = \frac{x}{20} + 6 \cdot \frac{20}{20}$

Step 6.2

Combine $6$ and $\frac{20}{20}$ .

${(y - 3)}^{2} = \frac{x}{20} + \frac{6 \cdot 20}{20}$

Step 6.3

Combine the numerators over the common denominator.

${(y - 3)}^{2} = \frac{x + 6 \cdot 20}{20}$

Step 6.4

Multiply $6$ by $20$ .

${(y - 3)}^{2} = \frac{x + 120}{20}$

Step 6.5

Reorder terms.

${(y - 3)}^{2} = \frac{1}{20} (x + 120)$