Trigonometry Examples

$\frac{x + 4}{2 x - 4} \cdot \frac{8 x^{2} - 32}{x^{2} - 16}$

Step 1

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

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{x + 4}{2 (x) - 4} \cdot \frac{8 x^{2} - 32}{x^{2} - 16}$

Step 1.2

Factor $2$ out of $- 4$ .

$\frac{x + 4}{2 x + 2 \cdot - 2} \cdot \frac{8 x^{2} - 32}{x^{2} - 16}$

Step 1.3

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

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 x^{2} - 32}{x^{2} - 16}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $8$ out of $8 x^{2} - 32$ .

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Step 2.1.1

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

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x^{2}) - 32}{x^{2} - 16}$

Step 2.1.2

Factor $8$ out of $- 32$ .

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 x^{2} + 8 \cdot - 4}{x^{2} - 16}$

Step 2.1.3

Factor $8$ out of $8 x^{2} + 8 \cdot - 4$ .

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x^{2} - 4)}{x^{2} - 16}$

Step 2.2

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

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x^{2} - 2^{2})}{x^{2} - 16}$

Step 2.3

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 $b = 2$ .

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x + 2) (x - 2)}{x^{2} - 16}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x + 2) (x - 2)}{x^{2} - 4^{2}}$

Step 3.2

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 $b = 4$ .

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x + 2) (x - 2)}{(x + 4) (x - 4)}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

Cancel the common factor.

$\frac{x + 4}{2 (x - 2)} \cdot \frac{8 (x + 2) (x - 2)}{(x + 4) (x - 4)}$

Step 4.1.2

Rewrite the expression.

$\frac{1}{2 (x - 2)} \cdot \frac{8 (x + 2) (x - 2)}{x - 4}$

Step 4.2

Cancel the common factor of $2 (x - 2)$ .

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Step 4.2.1

Factor $2 (x - 2)$ out of $8 (x + 2) (x - 2)$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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