Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 4$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.1

Move $x$ .

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

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $- 4$ by $2$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $- 1$ by $4$ .

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

Step 4.6

Subtract $x^{2}$ from $2 x^{2}$ .

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