Algebra Examples

$\frac{3}{a + 2} + \frac{4}{2 - a} - \frac{6}{a^{2} - 4}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{3}{a + 2} + \frac{4}{2 - a} - \frac{6}{a^{2} - 2^{2}}$

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 $a = a$ and $b = 2$ .

$\frac{3}{a + 2} + \frac{4}{2 - a} - \frac{6}{(a + 2) (a - 2)}$

Step 2

To write $\frac{3}{a + 2}$ as a fraction with a common denominator, multiply by $\frac{2 - a}{2 - a}$ .

$\frac{3}{a + 2} \cdot \frac{2 - a}{2 - a} + \frac{4}{2 - a} - \frac{6}{(a + 2) (a - 2)}$

Step 3

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

$\frac{3}{a + 2} \cdot \frac{2 - a}{2 - a} + \frac{4}{2 - a} \cdot \frac{a + 2}{a + 2} - \frac{6}{(a + 2) (a - 2)}$

Step 4

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

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Step 4.1

Multiply $\frac{3}{a + 2}$ by $\frac{2 - a}{2 - a}$ .

$\frac{3 (2 - a)}{(a + 2) (2 - a)} + \frac{4}{2 - a} \cdot \frac{a + 2}{a + 2} - \frac{6}{(a + 2) (a - 2)}$

Step 4.2

Multiply $\frac{4}{2 - a}$ by $\frac{a + 2}{a + 2}$ .

$\frac{3 (2 - a)}{(a + 2) (2 - a)} + \frac{4 (a + 2)}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 4.3

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

$\frac{3 (2 - a)}{(2 - a) (a + 2)} + \frac{4 (a + 2)}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 (2 - a) + 4 (a + 2)}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 \cdot 2 + 3 (- a) + 4 (a + 2)}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6.2

Multiply $3$ by $2$ .

$\frac{6 + 3 (- a) + 4 (a + 2)}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6.3

Multiply $- 1$ by $3$ .

$\frac{6 - 3 a + 4 (a + 2)}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6.4

Apply the distributive property.

$\frac{6 - 3 a + 4 a + 4 \cdot 2}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6.5

Multiply $4$ by $2$ .

$\frac{6 - 3 a + 4 a + 8}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6.6

Add $6$ and $8$ .

$\frac{- 3 a + 4 a + 14}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 6.7

Add $- 3 a$ and $4 a$ .

$\frac{a + 14}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (a - 2)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $a$ .

$\frac{a + 14}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (- 1 (- a) - 2)}$

Step 7.2

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

$\frac{a + 14}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (- 1 (- a) - 1 (2))}$

Step 7.3

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

$\frac{a + 14}{(2 - a) (a + 2)} - \frac{6}{(a + 2) (- 1 (- a + 2))}$

Step 7.4

Reorder terms.

$\frac{a + 14}{(- a + 2) (a + 2)} - \frac{6}{(a + 2) (- 1 (- a + 2))}$

Step 8

To write $\frac{a + 14}{(- a + 2) (a + 2)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{a + 14}{(- a + 2) (a + 2)} \cdot \frac{- 1}{- 1} - \frac{6}{(a + 2) (- 1 (- a + 2))}$

Step 9

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

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Step 9.1

Multiply $\frac{a + 14}{(- a + 2) (a + 2)}$ by $\frac{- 1}{- 1}$ .

$\frac{(a + 14) \cdot - 1}{(- a + 2) (a + 2) \cdot - 1} - \frac{6}{(a + 2) (- 1 (- a + 2))}$

Step 9.2

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

$\frac{(a + 14) \cdot - 1}{- (- a + 2) (a + 2)} - \frac{6}{(a + 2) (- 1 (- a + 2))}$

Step 9.3

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

$\frac{(a + 14) \cdot - 1}{- (- a + 2) (a + 2)} - \frac{6}{- (- a + 2) (a + 2)}$

Step 10

Combine the numerators over the common denominator.

$\frac{(a + 14) \cdot - 1 - 6}{- (- a + 2) (a + 2)}$

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

$\frac{a \cdot - 1 + 14 \cdot - 1 - 6}{- (- a + 2) (a + 2)}$

Step 11.2

Move $- 1$ to the left of $a$ .

$\frac{- 1 \cdot a + 14 \cdot - 1 - 6}{- (- a + 2) (a + 2)}$

Step 11.3

Multiply $14$ by $- 1$ .

$\frac{- 1 \cdot a - 14 - 6}{- (- a + 2) (a + 2)}$

Step 11.4

Rewrite $- 1 a$ as $- a$ .

$\frac{- a - 14 - 6}{- (- a + 2) (a + 2)}$

Step 11.5

Subtract $6$ from $- 14$ .

$\frac{- a - 20}{- (- a + 2) (a + 2)}$

Step 12

Simplify terms.

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Step 12.1

Move the negative in front of the fraction.

$- \frac{- a - 20}{(- a + 2) (a + 2)}$

Step 12.2

Factor $- 1$ out of $- a$ .

$- \frac{- (a) - 20}{(- a + 2) (a + 2)}$

Step 12.3

Rewrite $- 20$ as $- 1 (20)$ .

$- \frac{- (a) - 1 (20)}{(- a + 2) (a + 2)}$

Step 12.4

Factor $- 1$ out of $- (a) - 1 (20)$ .

$- \frac{- (a + 20)}{(- a + 2) (a + 2)}$

Step 12.5

Rewrite $- (a + 20)$ as $- 1 (a + 20)$ .

$- \frac{- 1 (a + 20)}{(- a + 2) (a + 2)}$

Step 12.6

Factor $- 1$ out of $- a$ .

$- \frac{- 1 (a + 20)}{(- (a) + 2) (a + 2)}$

Step 12.7

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

$- \frac{- 1 (a + 20)}{(- (a) - 1 (- 2)) (a + 2)}$

Step 12.8

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

$- \frac{- 1 (a + 20)}{- (a - 2) (a + 2)}$

Step 12.9

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

$- \frac{- 1 (a + 20)}{- 1 (a - 2) (a + 2)}$

Step 12.10

Cancel the common factor.

$- \frac{- 1 (a + 20)}{- 1 (a - 2) (a + 2)}$

Step 12.11

Rewrite the expression.

$- \frac{a + 20}{(a - 2) (a + 2)}$