Algebra Examples

$\frac{1}{x^{2} - 1} + \frac{2}{x^{2} + x}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{x^{2} - 1^{2}} + \frac{2}{x^{2} + x}$

Step 1.1.2

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 $b = 1$ .

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

Step 1.2

Factor $x$ out of $x^{2} + x$ .

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Step 1.2.1

Factor $x$ out of $x^{2}$ .

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

Step 1.2.2

Raise $x$ to the power of $1$ .

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

Step 1.2.3

Factor $x$ out of $x^{1}$ .

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

Step 1.2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 2

To write $\frac{1}{(x + 1) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3

To write $\frac{2}{x (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4

Write each expression with a common denominator of $(x + 1) (x - 1) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{(x + 1) (x - 1)}$ by $\frac{x}{x}$ .

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

Step 4.2

Multiply $\frac{2}{x (x + 1)}$ by $\frac{x - 1}{x - 1}$ .

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

Step 4.3

Reorder the factors of $(x + 1) (x - 1) x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $- 1$ .

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

Step 6.3

Add $x$ and $2 x$ .

$\frac{3 x - 2}{x (x + 1) (x - 1)}$