Algebra Examples

$\frac{3}{x^{2} - 25} - \frac{2}{x + 5}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{3}{x^{2} - 5^{2}} - \frac{2}{x + 5}$

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 $b = 5$ .

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

Step 2

To write $- \frac{2}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2}{x + 5}$ by $\frac{x - 5}{x - 5}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 2$ by $- 5$ .

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

Step 4.3

Add $3$ and $10$ .

$\frac{- 2 x + 13}{(x + 5) (x - 5)}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) + 13}{(x + 5) (x - 5)}$

Step 5.2

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (2 x) - 1 (- 13)}{(x + 5) (x - 5)}$

Step 5.3

Factor $- 1$ out of $- (2 x) - 1 (- 13)$ .

$\frac{- (2 x - 13)}{(x + 5) (x - 5)}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (2 x - 13)$ as $- 1 (2 x - 13)$ .

$\frac{- 1 (2 x - 13)}{(x + 5) (x - 5)}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{2 x - 13}{(x + 5) (x - 5)}$