Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $x^{2} - 4$ out of $(x^{2} - 4) x + (x - 2) (x^{2} - 4) \cdot 1$ .

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Step 4.1.1

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

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

Step 4.1.2

Factor $x^{2} - 4$ out of $(x - 2) (x^{2} - 4) \cdot 1$ .

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

Step 4.1.3

Factor $x^{2} - 4$ out of $(x^{2} - 4) (x) + (x^{2} - 4) ((x - 2) \cdot 1)$ .

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

Step 4.2

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

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

Step 4.3

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 $b = 2$ .

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

Step 4.4

Multiply $x - 2$ by $1$ .

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

Step 4.5

Add $x$ and $x$ .

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

Step 4.6

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

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Step 4.6.1

Factor $2$ out of $2 x$ .

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

Step 4.6.2

Factor $2$ out of $- 2$ .

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

Step 4.6.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x - 2$ out of $(x - 2) \cdot 3 + (x - 2) (x^{2} - 4) \cdot 1$ .

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Step 5.1.1

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

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

Step 5.1.2

Factor $x - 2$ out of $(x - 2) (x^{2} - 4) \cdot 1$ .

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

Step 5.1.3

Factor $x - 2$ out of $(x - 2) (3) + (x - 2) ((x^{2} - 4) \cdot 1)$ .

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

Step 5.2

Multiply $x^{2} - 4$ by $1$ .

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

Step 5.3

Subtract $4$ from $3$ .

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

Step 5.4

Rewrite $x^{2} - 1$ in a factored form.

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Step 5.4.1

Rewrite $1$ as $1^{2}$ .

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

Step 5.4.2

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 $b = 1$ .

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

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

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.

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

Step 6.1.2

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x - 1$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

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

$\frac{2 (x + 2)}{x + 1}$