Precalculus Examples

$x^{2} + 8 x = - 4 y - 8$

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

Rewrite the equation as $- 4 y - 8 = x^{2} + 8 x$ .

$- 4 y - 8 = x^{2} + 8 x$

Step 1.1.2

Add $8$ to both sides of the equation.

$- 4 y = x^{2} + 8 x + 8$

Step 1.1.3

Divide each term in $- 4 y = x^{2} + 8 x + 8$ by $- 4$ and simplify.

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Step 1.1.3.1

Divide each term in $- 4 y = x^{2} + 8 x + 8$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{x^{2}}{- 4} + \frac{8 x}{- 4} + \frac{8}{- 4}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Cancel the common factor of $- 4$ .

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Step 1.1.3.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{x^{2}}{- 4} + \frac{8 x}{- 4} + \frac{8}{- 4}$

Step 1.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{- 4} + \frac{8 x}{- 4} + \frac{8}{- 4}$

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.

$y = - \frac{x^{2}}{4} + \frac{8 x}{- 4} + \frac{8}{- 4}$

Step 1.1.3.3.1.2

Cancel the common factor of $8$ and $- 4$ .

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Step 1.1.3.3.1.2.1

Factor $4$ out of $8 x$ .

$y = - \frac{x^{2}}{4} + \frac{4 (2 x)}{- 4} + \frac{8}{- 4}$

Step 1.1.3.3.1.2.2

Move the negative one from the denominator of $\frac{2 x}{- 1}$ .

$y = - \frac{x^{2}}{4} - 1 \cdot (2 x) + \frac{8}{- 4}$

Step 1.1.3.3.1.3

Rewrite $- 1 \cdot (2 x)$ as $- (2 x)$ .

$y = - \frac{x^{2}}{4} - (2 x) + \frac{8}{- 4}$

Step 1.1.3.3.1.4

Multiply $2$ by $- 1$ .

$y = - \frac{x^{2}}{4} - 2 x + \frac{8}{- 4}$

Step 1.1.3.3.1.5

Divide $8$ by $- 4$ .

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

Step 1.2

Complete the square for $- \frac{x^{2}}{4} - 2 x - 2$ .

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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 $c$ .

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

$b = - 2$

$c = - 2$

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

$d = \frac{- 2}{2 (- \frac{1}{4})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 (- \frac{1}{4})}$

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.

$d = \frac{2 \cdot - 1}{2 (- \frac{1}{4})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.2

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 4$

Step 1.2.3.2.4

Multiply $4$ by $1$ .

$d = 4$

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

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

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

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

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 $4$ .

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

Step 1.2.4.2.1.2.2

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

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

Step 1.2.4.2.1.3

Divide $- 4$ by $4$ .

$e = - 2 - \frac{4}{- 1}$

Step 1.2.4.2.1.4

Divide $4$ by $- 1$ .

$e = - 2 - - 4$

Step 1.2.4.2.1.5

Multiply $- 1$ by $- 4$ .

$e = - 2 + 4$

Step 1.2.4.2.2

Add $- 2$ and $4$ .

$e = 2$

Step 1.2.5

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

$- \frac{1}{4} {(x + 4)}^{2} + 2$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = - 4$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- 4, 2)$

Step 4