Precalculus Examples

$x^{2} + 12 x + 2 y + 42 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

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

$12 x + 2 y + 42 = - x^{2}$

Step 1.1.1.2

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

$2 y + 42 = - x^{2} - 12 x$

Step 1.1.1.3

Subtract $42$ from both sides of the equation.

$2 y = - x^{2} - 12 x - 42$

Step 1.1.2

Divide each term in $2 y = - x^{2} - 12 x - 42$ by $2$ and simplify.

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Step 1.1.2.1

Divide each term in $2 y = - x^{2} - 12 x - 42$ by $2$ .

$\frac{2 y}{2} = \frac{- x^{2}}{2} + \frac{- 12 x}{2} + \frac{- 42}{2}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- x^{2}}{2} + \frac{- 12 x}{2} + \frac{- 42}{2}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{2} + \frac{- 12 x}{2} + \frac{- 42}{2}$

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

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{2} + \frac{- 12 x}{2} + \frac{- 42}{2}$

Step 1.1.2.3.1.2

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

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Step 1.1.2.3.1.2.1

Factor $2$ out of $- 12 x$ .

$y = - \frac{x^{2}}{2} + \frac{2 (- 6 x)}{2} + \frac{- 42}{2}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $2$ out of $2$ .

$y = - \frac{x^{2}}{2} + \frac{2 (- 6 x)}{2 (1)} + \frac{- 42}{2}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{2} + \frac{2 (- 6 x)}{2 \cdot 1} + \frac{- 42}{2}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{2} + \frac{- 6 x}{1} + \frac{- 42}{2}$

Step 1.1.2.3.1.2.2.4

Divide $- 6 x$ by $1$ .

$y = - \frac{x^{2}}{2} - 6 x + \frac{- 42}{2}$

Step 1.1.2.3.1.3

Divide $- 42$ by $2$ .

$y = - \frac{x^{2}}{2} - 6 x - 21$

Step 1.2

Complete the square for $- \frac{x^{2}}{2} - 6 x - 21$ .

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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 $c$ .

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

$b = - 6$

$c = - 21$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $- 6$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $- 6$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{- 3}{- \frac{1}{2}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 3 (- 1 \cdot 2)$

Step 1.2.3.2.3

Multiply $- 3 (- 1 \cdot 2)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $2$ .

$d = - 3 \cdot - 2$

Step 1.2.3.2.3.2

Multiply $- 3$ by $- 2$ .

$d = 6$

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

$e = - 21 - \frac{{(- 6)}^{2}}{4 (- \frac{1}{2})}$

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

Raise $- 6$ to the power of $2$ .

$e = - 21 - \frac{36}{4 (- \frac{1}{2})}$

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 $4$ .

$e = - 21 - \frac{36}{- 4 (\frac{1}{2})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

$e = - 21 - \frac{36}{\frac{- 4}{2}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $2$ .

$e = - 21 - \frac{36}{- 2}$

Step 1.2.4.2.1.4

Divide $36$ by $- 2$ .

$e = - 21 - - 18$

Step 1.2.4.2.1.5

Multiply $- 1$ by $- 18$ .

$e = - 21 + 18$

Step 1.2.4.2.2

Add $- 21$ and $18$ .

$e = - 3$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 6)}^{2} - 3$ .

$- \frac{1}{2} {(x + 6)}^{2} - 3$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + 6)}^{2} - 3$

Step 2

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

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

$h = - 6$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 6, - 3)$

Step 4