Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

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 $b = 4$ .

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

Step 2

Factor $x^{2} - 2 x - 8$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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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 $(x - 4) (x + 2)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor of $x - 4$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\frac{x + 4}{x}$