Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $x$ by $x$ .

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

Step 4.3

Move $- 2$ to the left of $x$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

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

Step 8.2

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

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Step 8.2.1

Move $x$ .

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

Step 8.2.2

Multiply $x$ by $x$ .

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

Step 8.3

Multiply $2$ by $- 1$ .

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

Step 8.4

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

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

Step 8.5

Add $- 2 x$ and $0$ .

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

Step 8.6

Subtract $2 x$ from $- 2 x$ .

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

Step 8.7

Factor $2$ out of $- 4 x + 2$ .

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Step 8.7.1

Factor $2$ out of $- 4 x$ .

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

Step 8.7.2

Factor $2$ out of $2$ .

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

Step 8.7.3

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

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

Step 9

Simplify with factoring out.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Simplify the expression.

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Step 9.4.1

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

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

Step 9.4.2

Move the negative in front of the fraction.

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