Pre-Algebra Examples

$\frac{4}{x + 5} + \frac{2 x}{x^{2} - 25}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{4}{x + 5} + \frac{2 x}{x^{2} - 5^{2}}$

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 $b = 5$ .

$\frac{4}{x + 5} + \frac{2 x}{(x + 5) (x - 5)}$

Step 2

To write $\frac{4}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{4}{x + 5} \cdot \frac{x - 5}{x - 5} + \frac{2 x}{(x + 5) (x - 5)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{4}{x + 5}$ by $\frac{x - 5}{x - 5}$ .

$\frac{4 (x - 5)}{(x + 5) (x - 5)} + \frac{2 x}{(x + 5) (x - 5)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 (x - 5) + 2 x}{(x + 5) (x - 5)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $2$ out of $4 (x - 5) + 2 x$ .

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Step 4.1.1

Factor $2$ out of $4 (x - 5)$ .

$\frac{2 (2 (x - 5)) + 2 x}{(x + 5) (x - 5)}$

Step 4.1.2

Factor $2$ out of $2 x$ .

$\frac{2 (2 (x - 5)) + 2 (x)}{(x + 5) (x - 5)}$

Step 4.1.3

Factor $2$ out of $2 (2 (x - 5)) + 2 (x)$ .

$\frac{2 (2 (x - 5) + x)}{(x + 5) (x - 5)}$

Step 4.2

Apply the distributive property.

$\frac{2 (2 x + 2 \cdot - 5 + x)}{(x + 5) (x - 5)}$

Step 4.3

Multiply $2$ by $- 5$ .

$\frac{2 (2 x - 10 + x)}{(x + 5) (x - 5)}$

Step 4.4

Add $2 x$ and $x$ .

$\frac{2 (3 x - 10)}{(x + 5) (x - 5)}$