Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

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

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Step 2.1

Factor $2$ out of $2 x$ .

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

Step 2.2

Factor $2$ out of $4$ .

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

Step 2.3

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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Step 3.1.1

Factor $2$ out of $2$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 4

Cancel the common factor of $x + 2$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $- 1$ to the left of $x$ .

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

Step 6.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.1.4

Multiply $x$ by $1$ .

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

Step 6.1.5

Multiply $- 1$ by $1$ .

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

Step 6.2

Add $- x$ and $x$ .

$(x^{2} + 0 - 1) \cdot \frac{2}{2 (1 + x) (1 - x)}$

Step 6.3

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

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $2$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

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

Step 7.2

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

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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 $b = 1$ .

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

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Cancel the common factor of $x + 1$ and $1 + x$ .

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Step 9.1.1

Reorder terms.

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

Step 9.1.2

Cancel the common factor.

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

Step 9.1.3

Rewrite the expression.

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

Step 9.2

Cancel the common factor of $x - 1$ and $1 - x$ .

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Step 9.2.1

Factor $- 1$ out of $x$ .

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

Step 9.2.2

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

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

Step 9.2.3

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

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

Step 9.2.4

Reorder terms.

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

Step 9.2.5

Cancel the common factor.

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

Step 9.2.6

Divide $- 1$ by $1$ .

$- 1$