Algebra Examples

$\frac{\frac{3}{x - 2} - \frac{6}{x^{2} - 4}}{\frac{3}{x + 2} + \frac{1}{x - 2}}$

Step 1

Multiply the numerator and denominator of the fraction by $(x - 2) (x^{2} - 4) (x + 2)$ .

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Step 1.1

Multiply $\frac{\frac{3}{x - 2} - \frac{6}{x^{2} - 4}}{\frac{3}{x + 2} + \frac{1}{x - 2}}$ by $\frac{(x - 2) (x^{2} - 4) (x + 2)}{(x - 2) (x^{2} - 4) (x + 2)}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

Factor $x - 2$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x^{2} - 4$ .

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Step 3.2.1

Move the leading negative in $- \frac{6}{x^{2} - 4}$ into the numerator.

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

Step 3.2.2

Factor $x^{2} - 4$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 2$ .

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Step 3.3.1

Factor $x + 2$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x - 2$ .

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Step 3.4.1

Factor $x - 2$ out of $(x - 2) (x^{2} - 4) (x + 2)$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $(x + 2) \cdot 3$ out of $(x^{2} - 4) (x + 2) \cdot 3 + (x - 2) (x + 2) \cdot - 6$ .

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Step 4.1.1

Factor $(x + 2) \cdot 3$ out of $(x^{2} - 4) (x + 2) \cdot 3$ .

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

Step 4.1.2

Factor $(x + 2) \cdot 3$ out of $(x - 2) (x + 2) \cdot - 6$ .

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

Step 4.1.3

Factor $(x + 2) \cdot 3$ out of $(x + 2) \cdot 3 (x^{2} - 4) + (x + 2) \cdot 3 ((x - 2) \cdot - 2)$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Move $- 2$ to the left of $x$ .

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

Step 4.4

Multiply $- 2$ by $- 2$ .

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

Step 4.5

Add $- 4$ and $4$ .

$\frac{(x + 2) \cdot 3 (x^{2} - 2 x + 0)}{(x - 2) (x^{2} - 4) \cdot 3 + (x^{2} - 4) (x + 2)}$

Step 4.6

Add $x^{2} - 2 x$ and $0$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x^{2} - 4$ out of $(x - 2) (x^{2} - 4) \cdot 3 + (x^{2} - 4) (x + 2)$ .

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Step 5.1.1

Factor $x^{2} - 4$ out of $(x - 2) (x^{2} - 4) \cdot 3$ .

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

Step 5.1.2

Factor $x^{2} - 4$ out of $(x^{2} - 4) ((x - 2) \cdot 3) + (x^{2} - 4) (x + 2)$ .

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

Step 5.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.3

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 $b = 2$ .

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) ((x - 2) \cdot 3 + x + 2)}$

Step 5.4

Apply the distributive property.

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) (x \cdot 3 - 2 \cdot 3 + x + 2)}$

Step 5.5

Move $3$ to the left of $x$ .

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) (3 \cdot x - 2 \cdot 3 + x + 2)}$

Step 5.6

Multiply $- 2$ by $3$ .

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

Step 5.7

Add $3 x$ and $x$ .

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

Step 5.8

Add $- 6$ and $2$ .

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

Step 5.9

Factor $4$ out of $4 x - 4$ .

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Step 5.9.1

Factor $4$ out of $4 x$ .

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

Step 5.9.2

Factor $4$ out of $- 4$ .

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) (4 (x) + 4 (- 1))}$

Step 5.9.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) (4 (x - 1))}$

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) \cdot 4 (x - 1)}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $x + 2$ .

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Step 6.1.1

Cancel the common factor.

$\frac{(x + 2) \cdot 3 x (x - 2)}{(x + 2) (x - 2) \cdot 4 (x - 1)}$

Step 6.1.2

Rewrite the expression.

$\frac{3 x (x - 2)}{(x - 2) \cdot 4 (x - 1)}$

Step 6.2

Cancel the common factor of $x - 2$ .

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Step 6.2.1

Cancel the common factor.

$\frac{3 x (x - 2)}{(x - 2) \cdot 4 (x - 1)}$

Step 6.2.2

Rewrite the expression.

$\frac{3 x}{4 (x - 1)}$