Algebra Examples

$\frac{6}{x^{2} - 4} + \frac{5}{x + 2} - \frac{5}{2 - x}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{6}{x^{2} - 2^{2}} + \frac{5}{x + 2} - \frac{5}{2 - x}$

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 $b = 2$ .

$\frac{6}{(x + 2) (x - 2)} + \frac{5}{x + 2} - \frac{5}{2 - x}$

Step 2

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

$\frac{6}{(x + 2) (x - 2)} + \frac{5}{x + 2} \cdot \frac{x - 2}{x - 2} - \frac{5}{2 - x}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{6}{(x + 2) (x - 2)} + \frac{5 (x - 2)}{(x + 2) (x - 2)} - \frac{5}{2 - x}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{6 + 5 (x - 2)}{(x + 2) (x - 2)} - \frac{5}{2 - x}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{6 + 5 x + 5 \cdot - 2}{(x + 2) (x - 2)} - \frac{5}{2 - x}$

Step 4.2

Multiply $5$ by $- 2$ .

$\frac{6 + 5 x - 10}{(x + 2) (x - 2)} - \frac{5}{2 - x}$

Step 4.3

Subtract $10$ from $6$ .

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

Step 5

Simplify with factoring out.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Reorder terms.

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

Step 6

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

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

Step 7

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

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

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

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

Step 10.2

Multiply $- 1$ by $5$ .

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

Step 10.3

Multiply $- 4$ by $- 1$ .

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

Step 10.4

Apply the distributive property.

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

Step 10.5

Multiply $- 5$ by $2$ .

$\frac{- 5 x + 4 - 5 x - 10}{- (x + 2) (x - 2)}$

Step 10.6

Subtract $5 x$ from $- 5 x$ .

$\frac{- 10 x + 4 - 10}{- (x + 2) (x - 2)}$

Step 10.7

Subtract $10$ from $4$ .

$\frac{- 10 x - 6}{- (x + 2) (x - 2)}$

Step 10.8

Factor $2$ out of $- 10 x - 6$ .

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Step 10.8.1

Factor $2$ out of $- 10 x$ .

$\frac{2 (- 5 x) - 6}{- (x + 2) (x - 2)}$

Step 10.8.2

Factor $2$ out of $- 6$ .

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

Step 10.8.3

Factor $2$ out of $2 (- 5 x) + 2 (- 3)$ .

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

Step 11

Simplify with factoring out.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

Factor $- 1$ out of $- 5 x$ .

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

Step 11.3

Rewrite $- 3$ as $- 1 (3)$ .

$- \frac{2 (- (5 x) - 1 (3))}{(x + 2) (x - 2)}$

Step 11.4

Factor $- 1$ out of $- (5 x) - 1 (3)$ .

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

Step 11.5

Simplify the expression.

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Step 11.5.1

Rewrite $- (5 x + 3)$ as $- 1 (5 x + 3)$ .

$- \frac{2 (- 1 (5 x + 3))}{(x + 2) (x - 2)}$

Step 11.5.2

Move the negative in front of the fraction.

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

Step 11.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{2 (5 x + 3)}{(x + 2) (x - 2)}$

Step 11.5.4

Multiply $\frac{2 (5 x + 3)}{(x + 2) (x - 2)}$ by $1$ .

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