Algebra Examples

$\frac{2 a}{2 a + 3} + \frac{5}{3 - 2 a} - \frac{4 a^{2} + 9}{4 a^{2} - 9}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

$\frac{2 a}{2 a + 3} + \frac{5}{3 - 2 a} - \frac{4 a^{2} + 9}{{(2 a)}^{2} - 9}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\frac{2 a}{2 a + 3} + \frac{5}{3 - 2 a} - \frac{4 a^{2} + 9}{{(2 a)}^{2} - 3^{2}}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a$ and $b = 3$ .

$\frac{2 a}{2 a + 3} + \frac{5}{3 - 2 a} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 2

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

$\frac{2 a}{2 a + 3} \cdot \frac{3 - 2 a}{3 - 2 a} + \frac{5}{3 - 2 a} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 3

To write $\frac{5}{3 - 2 a}$ as a fraction with a common denominator, multiply by $\frac{2 a + 3}{2 a + 3}$ .

$\frac{2 a}{2 a + 3} \cdot \frac{3 - 2 a}{3 - 2 a} + \frac{5}{3 - 2 a} \cdot \frac{2 a + 3}{2 a + 3} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 4

Write each expression with a common denominator of $(2 a + 3) (3 - 2 a)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

$\frac{2 a (3 - 2 a)}{(2 a + 3) (3 - 2 a)} + \frac{5}{3 - 2 a} \cdot \frac{2 a + 3}{2 a + 3} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 4.2

Multiply $\frac{5}{3 - 2 a}$ by $\frac{2 a + 3}{2 a + 3}$ .

$\frac{2 a (3 - 2 a)}{(2 a + 3) (3 - 2 a)} + \frac{5 (2 a + 3)}{(3 - 2 a) (2 a + 3)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 4.3

Reorder the factors of $(3 - 2 a) (2 a + 3)$ .

$\frac{2 a (3 - 2 a)}{(2 a + 3) (3 - 2 a)} + \frac{5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 a (3 - 2 a) + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 a \cdot 3 + 2 a (- 2 a) + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.2

Multiply $3$ by $2$ .

$\frac{6 a + 2 a (- 2 a) + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.3

Rewrite using the commutative property of multiplication.

$\frac{6 a + 2 \cdot - 2 a \cdot a + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.4

Simplify each term.

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Step 6.4.1

Multiply $a$ by $a$ by adding the exponents.

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Step 6.4.1.1

Move $a$ .

$\frac{6 a + 2 \cdot - 2 (a \cdot a) + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.4.1.2

Multiply $a$ by $a$ .

$\frac{6 a + 2 \cdot - 2 a^{2} + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.4.2

Multiply $2$ by $- 2$ .

$\frac{6 a - 4 a^{2} + 5 (2 a + 3)}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.5

Apply the distributive property.

$\frac{6 a - 4 a^{2} + 5 (2 a) + 5 \cdot 3}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.6

Multiply $2$ by $5$ .

$\frac{6 a - 4 a^{2} + 10 a + 5 \cdot 3}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.7

Multiply $5$ by $3$ .

$\frac{6 a - 4 a^{2} + 10 a + 15}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 6.8

Add $6 a$ and $10 a$ .

$\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (2 a - 3)}$

Step 7

Simplify with factoring out.

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Step 7.1

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

$\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a) - 3)}$

Step 7.2

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

$\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a) - 1 (3))}$

Step 7.3

Factor $- 1$ out of $- (- 2 a) - 1 (3)$ .

$\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (3 - 2 a)} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a + 3))}$

Step 7.4

Reorder terms.

$\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (- 2 a + 3)} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a + 3))}$

Step 8

To write $\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (- 2 a + 3)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (- 2 a + 3)} \cdot \frac{- 1}{- 1} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a + 3))}$

Step 9

Write each expression with a common denominator of $(2 a + 3) (- 2 a + 3) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{- 4 a^{2} + 16 a + 15}{(2 a + 3) (- 2 a + 3)}$ by $\frac{- 1}{- 1}$ .

$\frac{(- 4 a^{2} + 16 a + 15) \cdot - 1}{(2 a + 3) (- 2 a + 3) \cdot - 1} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a + 3))}$

Step 9.2

Reorder the factors of $(2 a + 3) (- 2 a + 3) \cdot - 1$ .

$\frac{(- 4 a^{2} + 16 a + 15) \cdot - 1}{- (- 2 a + 3) (2 a + 3)} - \frac{4 a^{2} + 9}{(2 a + 3) (- (- 2 a + 3))}$

Step 9.3

Reorder the factors of $(2 a + 3) (- (- 2 a + 3))$ .

$\frac{(- 4 a^{2} + 16 a + 15) \cdot - 1}{- (- 2 a + 3) (2 a + 3)} - \frac{4 a^{2} + 9}{- (- 2 a + 3) (2 a + 3)}$

Step 10

Combine the numerators over the common denominator.

$\frac{(- 4 a^{2} + 16 a + 15) \cdot - 1 - (4 a^{2} + 9)}{- (- 2 a + 3) (2 a + 3)}$

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

$\frac{- 4 a^{2} \cdot - 1 + 16 a \cdot - 1 + 15 \cdot - 1 - (4 a^{2} + 9)}{- (- 2 a + 3) (2 a + 3)}$

Step 11.2

Simplify.

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Step 11.2.1

Multiply $- 1$ by $- 4$ .

$\frac{4 a^{2} + 16 a \cdot - 1 + 15 \cdot - 1 - (4 a^{2} + 9)}{- (- 2 a + 3) (2 a + 3)}$

Step 11.2.2

Multiply $- 1$ by $16$ .

$\frac{4 a^{2} - 16 a + 15 \cdot - 1 - (4 a^{2} + 9)}{- (- 2 a + 3) (2 a + 3)}$

Step 11.2.3

Multiply $15$ by $- 1$ .

$\frac{4 a^{2} - 16 a - 15 - (4 a^{2} + 9)}{- (- 2 a + 3) (2 a + 3)}$

Step 11.3

Apply the distributive property.

$\frac{4 a^{2} - 16 a - 15 - (4 a^{2}) - 1 \cdot 9}{- (- 2 a + 3) (2 a + 3)}$

Step 11.4

Multiply $4$ by $- 1$ .

$\frac{4 a^{2} - 16 a - 15 - 4 a^{2} - 1 \cdot 9}{- (- 2 a + 3) (2 a + 3)}$

Step 11.5

Multiply $- 1$ by $9$ .

$\frac{4 a^{2} - 16 a - 15 - 4 a^{2} - 9}{- (- 2 a + 3) (2 a + 3)}$

Step 11.6

Subtract $4 a^{2}$ from $4 a^{2}$ .

$\frac{- 16 a - 15 + 0 - 9}{- (- 2 a + 3) (2 a + 3)}$

Step 11.7

Add $- 16 a$ and $0$ .

$\frac{- 16 a - 15 - 9}{- (- 2 a + 3) (2 a + 3)}$

Step 11.8

Subtract $9$ from $- 15$ .

$\frac{- 16 a - 24}{- (- 2 a + 3) (2 a + 3)}$

Step 11.9

Factor $8$ out of $- 16 a - 24$ .

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Step 11.9.1

Factor $8$ out of $- 16 a$ .

$\frac{8 (- 2 a) - 24}{- (- 2 a + 3) (2 a + 3)}$

Step 11.9.2

Factor $8$ out of $- 24$ .

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

Step 11.9.3

Factor $8$ out of $8 (- 2 a) + 8 (- 3)$ .

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

Step 12

Simplify terms.

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Step 12.1

Cancel the common factor of $- 2 a - 3$ and $2 a + 3$ .

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Step 12.1.1

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

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

Step 12.1.2

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

$\frac{8 (- (2 a) - 1 (3))}{- (- 2 a + 3) (2 a + 3)}$

Step 12.1.3

Factor $- 1$ out of $- (2 a) - 1 (3)$ .

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

Step 12.1.4

Rewrite $- (2 a + 3)$ as $- 1 (2 a + 3)$ .

$\frac{8 (- 1 (2 a + 3))}{- (- 2 a + 3) (2 a + 3)}$

Step 12.1.5

Cancel the common factor.

$\frac{8 (- 1 (2 a + 3))}{- (- 2 a + 3) (2 a + 3)}$

Step 12.1.6

Rewrite the expression.

$\frac{8 \cdot (- 1)}{- (- 2 a + 3)}$

Step 12.2

Dividing two negative values results in a positive value.

$\frac{8}{- 2 a + 3}$

Step 12.3

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

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

Step 12.4

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

$\frac{8}{- (2 a) - 1 (- 3)}$

Step 12.5

Factor $- 1$ out of $- (2 a) - 1 (- 3)$ .

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

Step 12.6

Rewrite negatives.

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Step 12.6.1

Rewrite $- (2 a - 3)$ as $- 1 (2 a - 3)$ .

$\frac{8}{- 1 (2 a - 3)}$

Step 12.6.2

Move the negative in front of the fraction.

$- \frac{8}{2 a - 3}$