Algebra Examples

$- 6 x^{2} - 36 x = y + 8$

Step 1

Complete the square.

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Step 1.1

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

$a = - 6$

$b = - 36$

$c = 0 = y + 8$

Step 1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 8$

Step 1.3

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

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Step 1.3.1

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

$d = \frac{- 36}{2 \cdot - 6}$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Cancel the common factor of $- 36$ and $2$ .

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Step 1.3.2.1.1

Factor $2$ out of $- 36$ .

$d = \frac{2 \cdot - 18}{2 \cdot - 6}$

Step 1.3.2.1.2

Cancel the common factors.

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Step 1.3.2.1.2.1

Factor $2$ out of $2 \cdot - 6$ .

$d = \frac{2 \cdot - 18}{2 (- 6)}$

Step 1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 18}{2 \cdot - 6}$

Step 1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 18}{- 6}$

Step 1.3.2.2

Cancel the common factor of $- 18$ and $- 6$ .

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Step 1.3.2.2.1

Factor $- 6$ out of $- 18$ .

$d = \frac{- 6 (3)}{- 6}$

Step 1.3.2.2.2

Cancel the common factors.

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Step 1.3.2.2.2.1

Factor $- 6$ out of $- 6$ .

$d = \frac{- 6 \cdot 3}{- 6 \cdot 1}$

Step 1.3.2.2.2.2

Cancel the common factor.

$d = \frac{- 6 \cdot 3}{- 6 \cdot 1}$

Step 1.3.2.2.2.3

Rewrite the expression.

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

Step 1.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

Step 1.4

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

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Step 1.4.1

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

$e = 0 - \frac{{(- 36)}^{2}}{4 \cdot - 6}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Raise $- 36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot - 6}$

Step 1.4.2.1.2

Multiply $4$ by $- 6$ .

$e = 0 - \frac{1296}{- 24}$

Step 1.4.2.1.3

Divide $1296$ by $- 24$ .

$e = 0 - - 54$

Step 1.4.2.1.4

Multiply $- 1$ by $- 54$ .

$e = 0 + 54$

Step 1.4.2.2

Add $0$ and $54$ .

$e = 54$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 6 {(x + 3)}^{2} + 54$ .

$- 6 {(x + 3)}^{2} + 54 = y + 8$

Step 2

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

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Step 2.1

Subtract $54$ from both sides of the equation.

$- 6 {(x + 3)}^{2} = y + 8 - 54$

Step 2.2

Subtract $54$ from $8$ .

$- 6 {(x + 3)}^{2} = y - 46$

Step 3

Divide each term in $- 6 {(x + 3)}^{2} = y - 46$ by $- 6$ and simplify.

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Step 3.1

Divide each term in $- 6 {(x + 3)}^{2} = y - 46$ by $- 6$ .

$\frac{- 6 {(x + 3)}^{2}}{- 6} = \frac{y}{- 6} + \frac{- 46}{- 6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 6 {(x + 3)}^{2}}{- 6} = \frac{y}{- 6} + \frac{- 46}{- 6}$

Step 3.2.1.2

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{y}{- 6} + \frac{- 46}{- 6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

${(x + 3)}^{2} = - \frac{y}{6} + \frac{- 46}{- 6}$

Step 3.3.1.2

Cancel the common factor of $- 46$ and $- 6$ .

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Step 3.3.1.2.1

Factor $- 2$ out of $- 46$ .

${(x + 3)}^{2} = - \frac{y}{6} + \frac{- 2 (23)}{- 6}$

Step 3.3.1.2.2

Cancel the common factors.

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Step 3.3.1.2.2.1

Factor $- 2$ out of $- 6$ .

${(x + 3)}^{2} = - \frac{y}{6} + \frac{- 2 \cdot 23}{- 2 \cdot 3}$

Step 3.3.1.2.2.2

Cancel the common factor.

${(x + 3)}^{2} = - \frac{y}{6} + \frac{- 2 \cdot 23}{- 2 \cdot 3}$

Step 3.3.1.2.2.3

Rewrite the expression.

${(x + 3)}^{2} = - \frac{y}{6} + \frac{23}{3}$

Step 4

Factor.

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Step 4.1

To write $\frac{23}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x + 3)}^{2} = - \frac{y}{6} + \frac{23}{3} \cdot \frac{2}{2}$

Step 4.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1

Multiply $\frac{23}{3}$ by $\frac{2}{2}$ .

${(x + 3)}^{2} = - \frac{y}{6} + \frac{23 \cdot 2}{3 \cdot 2}$

Step 4.2.2

Multiply $3$ by $2$ .

${(x + 3)}^{2} = - \frac{y}{6} + \frac{23 \cdot 2}{6}$

Step 4.3

Combine the numerators over the common denominator.

${(x + 3)}^{2} = \frac{- y + 23 \cdot 2}{6}$

Step 4.4

Multiply $23$ by $2$ .

${(x + 3)}^{2} = \frac{- y + 46}{6}$

Step 4.5

Factor $- 1$ out of $- y$ .

${(x + 3)}^{2} = \frac{- (y) + 46}{6}$

Step 4.6

Rewrite $46$ as $- 1 (- 46)$ .

${(x + 3)}^{2} = \frac{- (y) - 1 (- 46)}{6}$

Step 4.7

Factor $- 1$ out of $- (y) - 1 (- 46)$ .

${(x + 3)}^{2} = \frac{- (y - 46)}{6}$

Step 4.8

Move the negative in front of the fraction.

${(x + 3)}^{2} = - \frac{y - 46}{6}$

Step 4.9

Reorder terms.

${(x + 3)}^{2} = - (\frac{1}{6} (y - 46))$

Step 4.10

Remove parentheses.

${(x + 3)}^{2} = - \frac{1}{6} (y - 46)$