Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $x$ by $x$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

Multiply $- 2$ by $- 2$ .

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

Step 4.2.2

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

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

Step 4.3

Add $4$ and $1$ .

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} - \frac{x + 2}{2 - x}$

Step 5

Simplify with factoring out.

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Step 5.1

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

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} - \frac{x + 2}{- 1 (- 2) - x}$

Step 5.2

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

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} - \frac{x + 2}{- 1 (- 2) - (x)}$

Step 5.3

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

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} - \frac{x + 2}{- 1 (- 2 + x)}$

Step 5.4

Reorder terms.

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} - \frac{x + 2}{- 1 (x - 2)}$

Step 6

To write $\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} \cdot \frac{- 1}{- 1} - \frac{x + 2}{- 1 (x - 2)}$

Step 7

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

$\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)} \cdot \frac{- 1}{- 1} - \frac{x + 2}{- 1 (x - 2)} \cdot \frac{x + 2}{x + 2}$

Step 8

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

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Step 8.1

Multiply $\frac{x^{2} - 4 x + 5}{(x + 2) (x - 2)}$ by $\frac{- 1}{- 1}$ .

$\frac{(x^{2} - 4 x + 5) \cdot - 1}{(x + 2) (x - 2) \cdot - 1} - \frac{x + 2}{- 1 (x - 2)} \cdot \frac{x + 2}{x + 2}$

Step 8.2

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

$\frac{(x^{2} - 4 x + 5) \cdot - 1}{(x + 2) (x - 2) \cdot - 1} - \frac{(x + 2) (x + 2)}{- 1 (x - 2) (x + 2)}$

Step 8.3

Reorder the factors of $(x + 2) (x - 2) \cdot - 1$ .

$\frac{(x^{2} - 4 x + 5) \cdot - 1}{- (x + 2) (x - 2)} - \frac{(x + 2) (x + 2)}{- 1 (x - 2) (x + 2)}$

Step 8.4

Reorder the factors of $- 1 (x - 2) (x + 2)$ .

$\frac{(x^{2} - 4 x + 5) \cdot - 1}{- (x + 2) (x - 2)} - \frac{(x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 9

Combine the numerators over the common denominator.

$\frac{(x^{2} - 4 x + 5) \cdot - 1 - (x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{x^{2} \cdot - 1 - 4 x \cdot - 1 + 5 \cdot - 1 - (x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.2

Simplify.

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Step 10.2.1

Move $- 1$ to the left of $x^{2}$ .

$\frac{- 1 \cdot x^{2} - 4 x \cdot - 1 + 5 \cdot - 1 - (x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.2.2

Multiply $- 1$ by $- 4$ .

$\frac{- 1 \cdot x^{2} + 4 x + 5 \cdot - 1 - (x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.2.3

Multiply $5$ by $- 1$ .

$\frac{- 1 \cdot x^{2} + 4 x - 5 - (x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.3

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

$\frac{- x^{2} + 4 x - 5 - (x + 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.4

Apply the distributive property.

$\frac{- x^{2} + 4 x - 5 + (- x - 1 \cdot 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.5

Multiply $- 1$ by $2$ .

$\frac{- x^{2} + 4 x - 5 + (- x - 2) (x + 2)}{- (x + 2) (x - 2)}$

Step 10.6

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

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Step 10.6.1

Apply the distributive property.

$\frac{- x^{2} + 4 x - 5 - x (x + 2) - 2 (x + 2)}{- (x + 2) (x - 2)}$

Step 10.6.2

Apply the distributive property.

$\frac{- x^{2} + 4 x - 5 - x \cdot x - x \cdot 2 - 2 (x + 2)}{- (x + 2) (x - 2)}$

Step 10.6.3

Apply the distributive property.

$\frac{- x^{2} + 4 x - 5 - x \cdot x - x \cdot 2 - 2 x - 2 \cdot 2}{- (x + 2) (x - 2)}$

Step 10.7

Simplify and combine like terms.

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Step 10.7.1

Simplify each term.

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Step 10.7.1.1

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

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Step 10.7.1.1.1

Move $x$ .

$\frac{- x^{2} + 4 x - 5 - (x \cdot x) - x \cdot 2 - 2 x - 2 \cdot 2}{- (x + 2) (x - 2)}$

Step 10.7.1.1.2

Multiply $x$ by $x$ .

$\frac{- x^{2} + 4 x - 5 - x^{2} - x \cdot 2 - 2 x - 2 \cdot 2}{- (x + 2) (x - 2)}$

Step 10.7.1.2

Multiply $2$ by $- 1$ .

$\frac{- x^{2} + 4 x - 5 - x^{2} - 2 x - 2 x - 2 \cdot 2}{- (x + 2) (x - 2)}$

Step 10.7.1.3

Multiply $- 2$ by $2$ .

$\frac{- x^{2} + 4 x - 5 - x^{2} - 2 x - 2 x - 4}{- (x + 2) (x - 2)}$

Step 10.7.2

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

$\frac{- x^{2} + 4 x - 5 - x^{2} - 4 x - 4}{- (x + 2) (x - 2)}$

Step 10.8

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

$\frac{- 2 x^{2} + 4 x - 5 - 4 x - 4}{- (x + 2) (x - 2)}$

Step 10.9

Subtract $4 x$ from $4 x$ .

$\frac{- 2 x^{2} + 0 - 5 - 4}{- (x + 2) (x - 2)}$

Step 10.10

Add $- 2 x^{2}$ and $0$ .

$\frac{- 2 x^{2} - 5 - 4}{- (x + 2) (x - 2)}$

Step 10.11

Subtract $4$ from $- 5$ .

$\frac{- 2 x^{2} - 9}{- (x + 2) (x - 2)}$

Step 11

Simplify with factoring out.

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Step 11.1

Move the negative in front of the fraction.

$- \frac{- 2 x^{2} - 9}{(x + 2) (x - 2)}$

Step 11.2

Factor $- 1$ out of $- 2 x^{2}$ .

$- \frac{- (2 x^{2}) - 9}{(x + 2) (x - 2)}$

Step 11.3

Rewrite $- 9$ as $- 1 (9)$ .

$- \frac{- (2 x^{2}) - 1 (9)}{(x + 2) (x - 2)}$

Step 11.4

Factor $- 1$ out of $- (2 x^{2}) - 1 (9)$ .

$- \frac{- (2 x^{2} + 9)}{(x + 2) (x - 2)}$

Step 11.5

Simplify the expression.

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Step 11.5.1

Rewrite $- (2 x^{2} + 9)$ as $- 1 (2 x^{2} + 9)$ .

$- \frac{- 1 (2 x^{2} + 9)}{(x + 2) (x - 2)}$

Step 11.5.2

Move the negative in front of the fraction.

$- - \frac{2 x^{2} + 9}{(x + 2) (x - 2)}$

Step 11.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{2 x^{2} + 9}{(x + 2) (x - 2)}$

Step 11.5.4

Multiply $\frac{2 x^{2} + 9}{(x + 2) (x - 2)}$ by $1$ .

$\frac{2 x^{2} + 9}{(x + 2) (x - 2)}$