Precalculus Examples

$y^{2} + 6 y + 9 = 12 - 12 x$

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 $12 - 12 x = y^{2} + 6 y + 9$ .

$12 - 12 x = y^{2} + 6 y + 9$

Step 1.1.2

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

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Step 1.1.2.1

Subtract $12$ from both sides of the equation.

$- 12 x = y^{2} + 6 y + 9 - 12$

Step 1.1.2.2

Subtract $12$ from $9$ .

$- 12 x = y^{2} + 6 y - 3$

Step 1.1.3

Divide each term in $- 12 x = y^{2} + 6 y - 3$ by $- 12$ and simplify.

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Step 1.1.3.1

Divide each term in $- 12 x = y^{2} + 6 y - 3$ by $- 12$ .

$\frac{- 12 x}{- 12} = \frac{y^{2}}{- 12} + \frac{6 y}{- 12} + \frac{- 3}{- 12}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $- 12$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{- 12 x}{- 12} = \frac{y^{2}}{- 12} + \frac{6 y}{- 12} + \frac{- 3}{- 12}$

Step 1.1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{y^{2}}{- 12} + \frac{6 y}{- 12} + \frac{- 3}{- 12}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{12} + \frac{6 y}{- 12} + \frac{- 3}{- 12}$

Step 1.1.3.3.1.2

Cancel the common factor of $6$ and $- 12$ .

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Step 1.1.3.3.1.2.1

Factor $6$ out of $6 y$ .

$x = - \frac{y^{2}}{12} + \frac{6 (y)}{- 12} + \frac{- 3}{- 12}$

Step 1.1.3.3.1.2.2

Cancel the common factors.

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Step 1.1.3.3.1.2.2.1

Factor $6$ out of $- 12$ .

$x = - \frac{y^{2}}{12} + \frac{6 y}{6 \cdot - 2} + \frac{- 3}{- 12}$

Step 1.1.3.3.1.2.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{12} + \frac{6 y}{6 \cdot - 2} + \frac{- 3}{- 12}$

Step 1.1.3.3.1.2.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{12} + \frac{y}{- 2} + \frac{- 3}{- 12}$

Step 1.1.3.3.1.3

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{12} - \frac{y}{2} + \frac{- 3}{- 12}$

Step 1.1.3.3.1.4

Cancel the common factor of $- 3$ and $- 12$ .

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Step 1.1.3.3.1.4.1

Factor $- 3$ out of $- 3$ .

$x = - \frac{y^{2}}{12} - \frac{y}{2} + \frac{- 3 (1)}{- 12}$

Step 1.1.3.3.1.4.2

Cancel the common factors.

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Step 1.1.3.3.1.4.2.1

Factor $- 3$ out of $- 12$ .

$x = - \frac{y^{2}}{12} - \frac{y}{2} + \frac{- 3 \cdot 1}{- 3 \cdot 4}$

Step 1.1.3.3.1.4.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{12} - \frac{y}{2} + \frac{- 3 \cdot 1}{- 3 \cdot 4}$

Step 1.1.3.3.1.4.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{12} - \frac{y}{2} + \frac{1}{4}$

Step 1.2

Complete the square for $- \frac{y^{2}}{12} - \frac{y}{2} + \frac{1}{4}$ .

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Step 1.2.1

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

$a = - \frac{1}{12}$

$b = - \frac{1}{2}$

$c = \frac{1}{4}$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.3

Combine $2$ and $\frac{1}{12}$ .

$d = \frac{1}{2} \cdot \frac{1}{\frac{2}{12}}$

Step 1.2.3.2.4

Cancel the common factor of $2$ and $12$ .

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $12$ .

$d = \frac{1}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Step 1.2.3.2.4.2.2

Cancel the common factor.

$d = \frac{1}{2} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Step 1.2.3.2.4.2.3

Rewrite the expression.

$d = \frac{1}{2} \cdot \frac{1}{\frac{1}{6}}$

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.6

Multiply $6$ by $1$ .

$d = \frac{1}{2} \cdot 6$

Step 1.2.3.2.7

Cancel the common factor of $2$ .

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Step 1.2.3.2.7.1

Factor $2$ out of $6$ .

$d = \frac{1}{2} \cdot (2 (3))$

Step 1.2.3.2.7.2

Cancel the common factor.

$d = \frac{1}{2} \cdot (2 \cdot 3)$

Step 1.2.3.2.7.3

Rewrite the expression.

$d = 3$

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 1.2.4.2.1.1.2

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

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

Step 1.2.4.2.1.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.2.4.2.1.1.4

One to any power is one.

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

Step 1.2.4.2.1.1.5

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

$e = \frac{1}{4} - \frac{1 (\frac{1}{4})}{4 (- \frac{1}{12})}$

Step 1.2.4.2.1.1.6

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

$e = \frac{1}{4} - \frac{\frac{1}{4}}{4 (- \frac{1}{12})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{- 4 (\frac{1}{12})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{12}$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{- 4}{12}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

Cancel the common factor of $- 4$ and $12$ .

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Step 1.2.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{4 (- 1)}{12}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{- 1}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = \frac{1}{4} - \frac{\frac{1}{4}}{- \frac{1}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{4} - (\frac{1}{4} (- 1 \cdot 3))$

Step 1.2.4.2.1.5

Multiply $- 1$ by $3$ .

$e = \frac{1}{4} - (\frac{1}{4} \cdot - 3)$

Step 1.2.4.2.1.6

Combine $\frac{1}{4}$ and $- 3$ .

$e = \frac{1}{4} - \frac{- 3}{4}$

Step 1.2.4.2.1.7

Move the negative in front of the fraction.

$e = \frac{1}{4} - - \frac{3}{4}$

Step 1.2.4.2.1.8

Multiply $- - \frac{3}{4}$ .

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Step 1.2.4.2.1.8.1

Multiply $- 1$ by $- 1$ .

$e = \frac{1}{4} + 1 (\frac{3}{4})$

Step 1.2.4.2.1.8.2

Multiply $\frac{3}{4}$ by $1$ .

$e = \frac{1}{4} + \frac{3}{4}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{1 + 3}{4}$

Step 1.2.4.2.3

Add $1$ and $3$ .

$e = \frac{4}{4}$

Step 1.2.4.2.4

Divide $4$ by $4$ .

$e = 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(y + 3)}^{2} + 1$ .

$- \frac{1}{12} {(y + 3)}^{2} + 1$

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{12} \cdot {(y + 3)}^{2} + 1$

Step 2

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

$a = - \frac{1}{12}$

$h = 1$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(1, - 3)$

Step 4