Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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 $b = 2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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 $b = 1$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Factor $x$ out of $2 x$ .

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

Step 3.1.3

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

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

Step 3.2

Cancel the common factor of $x + 2$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 1$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\frac{x - 2}{x - 1} \cdot \frac{1}{x}$

Step 3.4

Multiply $\frac{x - 2}{x - 1}$ by $\frac{1}{x}$ .

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

Step 3.5

Reorder factors in $\frac{x - 2}{(x - 1) x}$ .

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