Algebra Examples

$\frac{2 x^{2} + 8 x}{x} = \frac{3 x^{2} - 108}{x - 6}$

Step 1

Factor each term.

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Step 1.1

Factor $2 x$ out of $2 x^{2} + 8 x$ .

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Step 1.1.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{2 x (x) + 8 x}{x} = \frac{3 x^{2} - 108}{x - 6}$

Step 1.1.2

Factor $2 x$ out of $8 x$ .

$\frac{2 x (x) + 2 x (4)}{x} = \frac{3 x^{2} - 108}{x - 6}$

Step 1.1.3

Factor $2 x$ out of $2 x (x) + 2 x (4)$ .

$\frac{2 x (x + 4)}{x} = \frac{3 x^{2} - 108}{x - 6}$

Step 1.2

Reduce the expression by cancelling the common factors.

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Step 1.2.1

Reduce the expression $\frac{2 x (x + 4)}{x}$ by cancelling the common factors.

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 x (x + 4)}{x} = \frac{3 x^{2} - 108}{x - 6}$

Step 1.2.1.2

Rewrite the expression.

$\frac{2 (x + 4)}{1} = \frac{3 x^{2} - 108}{x - 6}$

Step 1.2.2

Divide $2 (x + 4)$ by $1$ .

$2 (x + 4) = \frac{3 x^{2} - 108}{x - 6}$

Step 1.3

Factor $3$ out of $3 x^{2} - 108$ .

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Step 1.3.1

Factor $3$ out of $3 x^{2}$ .

$2 (x + 4) = \frac{3 (x^{2}) - 108}{x - 6}$

Step 1.3.2

Factor $3$ out of $- 108$ .

$2 (x + 4) = \frac{3 x^{2} + 3 \cdot - 36}{x - 6}$

Step 1.3.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 36$ .

$2 (x + 4) = \frac{3 (x^{2} - 36)}{x - 6}$

Step 1.4

Rewrite $36$ as $6^{2}$ .

$2 (x + 4) = \frac{3 (x^{2} - 6^{2})}{x - 6}$

Step 1.5

Factor.

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Step 1.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$2 (x + 4) = \frac{3 ((x + 6) (x - 6))}{x - 6}$

Step 1.5.2

Remove unnecessary parentheses.

$2 (x + 4) = \frac{3 (x + 6) (x - 6)}{x - 6}$

Step 1.6

Reduce the expression by cancelling the common factors.

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Step 1.6.1

Reduce the expression $\frac{3 (x + 6) (x - 6)}{x - 6}$ by cancelling the common factors.

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Step 1.6.1.1

Cancel the common factor.

$2 (x + 4) = \frac{3 (x + 6) (x - 6)}{x - 6}$

Step 1.6.1.2

Rewrite the expression.

$2 (x + 4) = \frac{3 (x + 6)}{1}$

Step 1.6.2

Divide $3 (x + 6)$ by $1$ .

$2 (x + 4) = 3 (x + 6)$

Step 2

Solve the equation.

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Step 2.1

Simplify $2 (x + 4)$ .

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Step 2.1.1

Rewrite.

$0 + 0 + 2 (x + 4) = 3 (x + 6)$

Step 2.1.2

Simplify by adding zeros.

$2 (x + 4) = 3 (x + 6)$

Step 2.1.3

Apply the distributive property.

$2 x + 2 \cdot 4 = 3 (x + 6)$

Step 2.1.4

Multiply $2$ by $4$ .

$2 x + 8 = 3 (x + 6)$

Step 2.2

Simplify $3 (x + 6)$ .

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Step 2.2.1

Apply the distributive property.

$2 x + 8 = 3 x + 3 \cdot 6$

Step 2.2.2

Multiply $3$ by $6$ .

$2 x + 8 = 3 x + 18$

Step 2.3

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.1

Subtract $3 x$ from both sides of the equation.

$2 x + 8 - 3 x = 18$

Step 2.3.2

Subtract $3 x$ from $2 x$ .

$- x + 8 = 18$

Step 2.4

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

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Step 2.4.1

Subtract $8$ from both sides of the equation.

$- x = 18 - 8$

Step 2.4.2

Subtract $8$ from $18$ .

$- x = 10$

Step 2.5

Divide each term in $- x = 10$ by $- 1$ and simplify.

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Step 2.5.1

Divide each term in $- x = 10$ by $- 1$ .

$\frac{- x}{- 1} = \frac{10}{- 1}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{10}{- 1}$

Step 2.5.2.2

Divide $x$ by $1$ .

$x = \frac{10}{- 1}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $10$ by $- 1$ .

$x = - 10$