Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 4$ .

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

Step 2

To write $\frac{2}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $2$ out of $16 + 2 (x - 4)$ .

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Step 4.1.1

Factor $2$ out of $16$ .

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

Step 4.1.2

Factor $2$ out of $2 \cdot 8 + 2 (x - 4)$ .

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

Step 4.2

Subtract $4$ from $8$ .

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

Step 5

Cancel the common factor of $x + 4$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{2}{x - 4}$