Precalculus Examples

$y^{2} - 5 x + 12 y = - 16$

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

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

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Step 1.1.1.1

Subtract $y^{2}$ from both sides of the equation.

$- 5 x + 12 y = - 16 - y^{2}$

Step 1.1.1.2

Subtract $12 y$ from both sides of the equation.

$- 5 x = - 16 - y^{2} - 12 y$

Step 1.1.2

Divide each term in $- 5 x = - 16 - y^{2} - 12 y$ by $- 5$ and simplify.

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Step 1.1.2.1

Divide each term in $- 5 x = - 16 - y^{2} - 12 y$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 16}{- 5} + \frac{- y^{2}}{- 5} + \frac{- 12 y}{- 5}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 16}{- 5} + \frac{- y^{2}}{- 5} + \frac{- 12 y}{- 5}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 5} + \frac{- y^{2}}{- 5} + \frac{- 12 y}{- 5}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{16}{5} + \frac{- y^{2}}{- 5} + \frac{- 12 y}{- 5}$

Step 1.1.2.3.1.2

Dividing two negative values results in a positive value.

$x = \frac{16}{5} + \frac{y^{2}}{5} + \frac{- 12 y}{- 5}$

Step 1.1.2.3.1.3

Dividing two negative values results in a positive value.

$x = \frac{16}{5} + \frac{y^{2}}{5} + \frac{12 y}{5}$

Step 1.2

Complete the square for $\frac{16}{5} + \frac{y^{2}}{5} + \frac{12 y}{5}$ .

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Step 1.2.1

Move $\frac{16}{5}$ .

$\frac{y^{2}}{5} + \frac{12 y}{5} + \frac{16}{5}$

Step 1.2.2

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

$a = \frac{1}{5}$

$b = \frac{12}{5}$

$c = \frac{16}{5}$

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 $d = \frac{b}{2 a}$ .

$d = \frac{\frac{12}{5}}{2 (\frac{1}{5})}$

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.

$d = \frac{12}{5} \cdot \frac{1}{2 (\frac{1}{5})}$

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Factor $2$ out of $12$ .

$d = \frac{2 (6)}{5} \cdot \frac{1}{2 (\frac{1}{5})}$

Step 1.2.4.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 6}{5} \cdot \frac{1}{2 (\frac{1}{5})}$

Step 1.2.4.2.2.3

Rewrite the expression.

$d = \frac{6}{5} \cdot \frac{1}{\frac{1}{5}}$

Step 1.2.4.2.3

Multiply $\frac{6}{5}$ by $\frac{1}{\frac{1}{5}}$ .

$d = \frac{6}{5 (\frac{1}{5})}$

Step 1.2.4.2.4

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

$d = \frac{6}{\frac{5}{5}}$

Step 1.2.4.2.5

Cancel the common factor of $5$ .

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Step 1.2.4.2.5.1

Cancel the common factor.

$d = \frac{6}{\frac{5}{5}}$

Step 1.2.4.2.5.2

Rewrite the expression.

$d = \frac{6}{1}$

Step 1.2.4.2.6

Divide $6$ by $1$ .

$d = 6$

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

$e = \frac{16}{5} - \frac{{(\frac{12}{5})}^{2}}{4 (\frac{1}{5})}$

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 $\frac{12}{5}$ .

$e = \frac{16}{5} - \frac{\frac{12^{2}}{5^{2}}}{4 (\frac{1}{5})}$

Step 1.2.5.2.1.1.2

Raise $12$ to the power of $2$ .

$e = \frac{16}{5} - \frac{\frac{144}{5^{2}}}{4 (\frac{1}{5})}$

Step 1.2.5.2.1.1.3

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

$e = \frac{16}{5} - \frac{\frac{144}{25}}{4 (\frac{1}{5})}$

Step 1.2.5.2.1.2

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

$e = \frac{16}{5} - \frac{\frac{144}{25}}{\frac{4}{5}}$

Step 1.2.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{16}{5} - (\frac{144}{25} \cdot \frac{5}{4})$

Step 1.2.5.2.1.4

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.4.1

Factor $4$ out of $144$ .

$e = \frac{16}{5} - (\frac{4 (36)}{25} \cdot \frac{5}{4})$

Step 1.2.5.2.1.4.2

Cancel the common factor.

$e = \frac{16}{5} - (\frac{4 \cdot 36}{25} \cdot \frac{5}{4})$

Step 1.2.5.2.1.4.3

Rewrite the expression.

$e = \frac{16}{5} - (\frac{36}{25} \cdot 5)$

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 $25$ .

$e = \frac{16}{5} - (\frac{36}{5 (5)} \cdot 5)$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = \frac{16}{5} - (\frac{36}{5 \cdot 5} \cdot 5)$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = \frac{16}{5} - \frac{36}{5}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{16 - 36}{5}$

Step 1.2.5.2.3

Subtract $36$ from $16$ .

$e = \frac{- 20}{5}$

Step 1.2.5.2.4

Divide $- 20$ by $5$ .

$e = - 4$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{5} {(y + 6)}^{2} - 4$ .

$\frac{1}{5} {(y + 6)}^{2} - 4$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{5} \cdot {(y + 6)}^{2} - 4$

Step 2

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

$a = \frac{1}{5}$

$h = - 4$

$k = - 6$

Step 3

Find the vertex $(h, k)$ .

$(- 4, - 6)$

Step 4