Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 5$ .

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Reorder the factors of $(x - 5) (x + 5)$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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Step 4.2.1

Move $x$ .

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

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $- 5$ by $- 1$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $2$ by $5$ .

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

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $x^{2}$ from $3 x^{2}$ .

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

Step 5.2

Add $5 x$ and $2 x$ .

$\frac{2 x^{2} + 7 x + 10}{(x + 5) (x - 5)}$