Algebra Examples

$\frac{x}{x^{2} - 4} + \frac{1}{x}$

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x + 2) (x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x + 2) (x - 2) x$ .

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

Step 4.4

Reorder the factors of $x ((x + 2) (x - 2))$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $x$ by $x$ .

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

Step 6.2

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

Apply the distributive property.

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

Step 6.2.3

Apply the distributive property.

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

Step 6.3

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 6.3.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Step 6.3.2

Add $- 2 x$ and $2 x$ .

$\frac{x^{2} + x \cdot x + 0 + 2 \cdot - 2}{x (x + 2) (x - 2)}$

Step 6.3.3

Add $x \cdot x$ and $0$ .

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

Step 6.4

Simplify each term.

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Step 6.4.1

Multiply $x$ by $x$ .

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

Step 6.4.2

Multiply $2$ by $- 2$ .

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

Step 6.5

Add $x^{2}$ and $x^{2}$ .

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

Step 6.6

Factor $2$ out of $2 x^{2} - 4$ .

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Step 6.6.1

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

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

Step 6.6.2

Factor $2$ out of $- 4$ .

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

Step 6.6.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 2$ .

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