Precalculus Examples

${(y - 4)}^{2} = 20 (x + 2)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

Rewrite the equation as $20 (x + 2) = {(y - 4)}^{2}$ .

$20 (x + 2) = {(y - 4)}^{2}$

Step 1.1.2

Divide each term in $20 (x + 2) = {(y - 4)}^{2}$ by $20$ and simplify.

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Step 1.1.2.1

Divide each term in $20 (x + 2) = {(y - 4)}^{2}$ by $20$ .

$\frac{20 (x + 2)}{20} = \frac{{(y - 4)}^{2}}{20}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $20$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{20 (x + 2)}{20} = \frac{{(y - 4)}^{2}}{20}$

Step 1.1.2.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{{(y - 4)}^{2}}{20}$

Step 1.1.3

Subtract $2$ from both sides of the equation.

$x = \frac{{(y - 4)}^{2}}{20} - 2$

Step 1.1.4

Reorder terms.

$x = \frac{1}{20} \cdot {(y - 4)}^{2} - 2$

Step 1.2

Complete the square for $\frac{1}{20} \cdot {(y - 4)}^{2} - 2$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(y - 4)}^{2}$ as $(y - 4) (y - 4)$ .

$\frac{1}{20} \cdot ((y - 4) (y - 4)) - 2$

Step 1.2.1.1.2

Expand $(y - 4) (y - 4)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{20} \cdot (y (y - 4) - 4 (y - 4)) - 2$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{20} \cdot (y \cdot y + y \cdot - 4 - 4 (y - 4)) - 2$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{20} \cdot (y \cdot y + y \cdot - 4 - 4 y - 4 \cdot - 4) - 2$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $y$ by $y$ .

$\frac{1}{20} \cdot (y^{2} + y \cdot - 4 - 4 y - 4 \cdot - 4) - 2$

Step 1.2.1.1.3.1.2

Move $- 4$ to the left of $y$ .

$\frac{1}{20} \cdot (y^{2} - 4 \cdot y - 4 y - 4 \cdot - 4) - 2$

Step 1.2.1.1.3.1.3

Multiply $- 4$ by $- 4$ .

$\frac{1}{20} \cdot (y^{2} - 4 y - 4 y + 16) - 2$

Step 1.2.1.1.3.2

Subtract $4 y$ from $- 4 y$ .

$\frac{1}{20} \cdot (y^{2} - 8 y + 16) - 2$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{20} y^{2} + \frac{1}{20} (- 8 y) + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $\frac{1}{20}$ and $y^{2}$ .

$\frac{y^{2}}{20} + \frac{1}{20} (- 8 y) + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.2

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.2.1

Factor $4$ out of $20$ .

$\frac{y^{2}}{20} + \frac{1}{4 (5)} (- 8 y) + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.2.2

Factor $4$ out of $- 8 y$ .

$\frac{y^{2}}{20} + \frac{1}{4 (5)} (4 (- 2 y)) + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.2.3

Cancel the common factor.

$\frac{y^{2}}{20} + \frac{1}{4 \cdot 5} (4 (- 2 y)) + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{y^{2}}{20} + \frac{1}{5} (- 2 y) + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.3

Combine $- 2$ and $\frac{1}{5}$ .

$\frac{y^{2}}{20} + \frac{- 2}{5} y + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.4

Combine $\frac{- 2}{5}$ and $y$ .

$\frac{y^{2}}{20} + \frac{- 2 y}{5} + \frac{1}{20} \cdot 16 - 2$

Step 1.2.1.1.5.5

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.5.1

Factor $4$ out of $20$ .

$\frac{y^{2}}{20} + \frac{- 2 y}{5} + \frac{1}{4 (5)} \cdot 16 - 2$

Step 1.2.1.1.5.5.2

Factor $4$ out of $16$ .

$\frac{y^{2}}{20} + \frac{- 2 y}{5} + \frac{1}{4 \cdot 5} \cdot (4 \cdot 4) - 2$

Step 1.2.1.1.5.5.3

Cancel the common factor.

$\frac{y^{2}}{20} + \frac{- 2 y}{5} + \frac{1}{4 \cdot 5} \cdot (4 \cdot 4) - 2$

Step 1.2.1.1.5.5.4

Rewrite the expression.

$\frac{y^{2}}{20} + \frac{- 2 y}{5} + \frac{1}{5} \cdot 4 - 2$

Step 1.2.1.1.5.6

Combine $\frac{1}{5}$ and $4$ .

$\frac{y^{2}}{20} + \frac{- 2 y}{5} + \frac{4}{5} - 2$

Step 1.2.1.1.6

Move the negative in front of the fraction.

$\frac{y^{2}}{20} - \frac{2 y}{5} + \frac{4}{5} - 2$

Step 1.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{y^{2}}{20} - \frac{2 y}{5} + \frac{4}{5} - 2 \cdot \frac{5}{5}$

Step 1.2.1.3

Combine $- 2$ and $\frac{5}{5}$ .

$\frac{y^{2}}{20} - \frac{2 y}{5} + \frac{4}{5} + \frac{- 2 \cdot 5}{5}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{y^{2}}{20} - \frac{2 y}{5} + \frac{4 - 2 \cdot 5}{5}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $- 2$ by $5$ .

$\frac{y^{2}}{20} - \frac{2 y}{5} + \frac{4 - 10}{5}$

Step 1.2.1.5.2

Subtract $10$ from $4$ .

$\frac{y^{2}}{20} - \frac{2 y}{5} + \frac{- 6}{5}$

Step 1.2.1.6

Move the negative in front of the fraction.

$\frac{y^{2}}{20} - \frac{2 y}{5} - \frac{6}{5}$

Step 1.2.2

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

$a = \frac{1}{20}$

$b = - \frac{2}{5}$

$c = - \frac{6}{5}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{- \frac{2}{5}}{2 (\frac{1}{20})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{2}{5} \cdot \frac{1}{2 (\frac{1}{20})}$

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$d = \frac{- 2}{5} \cdot \frac{1}{2 (\frac{1}{20})}$

Step 1.2.4.2.2.2

Factor $2$ out of $- 2$ .

$d = \frac{2 (- 1)}{5} \cdot \frac{1}{2 (\frac{1}{20})}$

Step 1.2.4.2.2.3

Cancel the common factor.

$d = \frac{2 \cdot - 1}{5} \cdot \frac{1}{2 (\frac{1}{20})}$

Step 1.2.4.2.2.4

Rewrite the expression.

$d = \frac{- 1}{5} \cdot \frac{1}{\frac{1}{20}}$

Step 1.2.4.2.3

Multiply $\frac{- 1}{5}$ by $\frac{1}{\frac{1}{20}}$ .

$d = \frac{- 1}{5 (\frac{1}{20})}$

Step 1.2.4.2.4

Combine $5$ and $\frac{1}{20}$ .

$d = \frac{- 1}{\frac{5}{20}}$

Step 1.2.4.2.5

Cancel the common factor of $5$ and $20$ .

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Step 1.2.4.2.5.1

Factor $5$ out of $5$ .

$d = \frac{- 1}{\frac{5 (1)}{20}}$

Step 1.2.4.2.5.2

Cancel the common factors.

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Step 1.2.4.2.5.2.1

Factor $5$ out of $20$ .

$d = \frac{- 1}{\frac{5 \cdot 1}{5 \cdot 4}}$

Step 1.2.4.2.5.2.2

Cancel the common factor.

$d = \frac{- 1}{\frac{5 \cdot 1}{5 \cdot 4}}$

Step 1.2.4.2.5.2.3

Rewrite the expression.

$d = \frac{- 1}{\frac{1}{4}}$

Step 1.2.4.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

Step 1.2.4.2.7

Multiply $- 1$ by $4$ .

$d = - 4$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{6}{5} - \frac{{(- \frac{2}{5})}^{2}}{4 (\frac{1}{20})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $- \frac{2}{5}$ .

$e = - \frac{6}{5} - \frac{{(- 1)}^{2} {(\frac{2}{5})}^{2}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.2

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

$e = - \frac{6}{5} - \frac{1 {(\frac{2}{5})}^{2}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.3

Apply the product rule to $\frac{2}{5}$ .

$e = - \frac{6}{5} - \frac{1 \frac{2^{2}}{5^{2}}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.4

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

$e = - \frac{6}{5} - \frac{1 \frac{4}{5^{2}}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.5

Raise $5$ to the power of $2$ .

$e = - \frac{6}{5} - \frac{1 (\frac{4}{25})}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.6

Multiply $\frac{4}{25}$ by $1$ .

$e = - \frac{6}{5} - \frac{\frac{4}{25}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.2

Combine $4$ and $\frac{1}{20}$ .

$e = - \frac{6}{5} - \frac{\frac{4}{25}}{\frac{4}{20}}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ and $20$ .

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Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = - \frac{6}{5} - \frac{\frac{4}{25}}{\frac{4 (1)}{20}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $4$ out of $20$ .

$e = - \frac{6}{5} - \frac{\frac{4}{25}}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = - \frac{6}{5} - \frac{\frac{4}{25}}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = - \frac{6}{5} - \frac{\frac{4}{25}}{\frac{1}{5}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{6}{5} - (\frac{4}{25} \cdot 5)$

Step 1.2.5.2.1.5

Cancel the common factor of $5$ .

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Step 1.2.5.2.1.5.1

Factor $5$ out of $25$ .

$e = - \frac{6}{5} - (\frac{4}{5 (5)} \cdot 5)$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = - \frac{6}{5} - (\frac{4}{5 \cdot 5} \cdot 5)$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = - \frac{6}{5} - \frac{4}{5}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 6 - 4}{5}$

Step 1.2.5.2.3

Subtract $4$ from $- 6$ .

$e = \frac{- 10}{5}$

Step 1.2.5.2.4

Divide $- 10$ by $5$ .

$e = - 2$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{20} {(y - 4)}^{2} - 2$ .

$\frac{1}{20} {(y - 4)}^{2} - 2$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{20} \cdot {(y - 4)}^{2} - 2$

Step 2

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = \frac{1}{20}$

$h = - 2$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 4)$

Step 4