Finite Math Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 2 b$ .

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

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

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Step 7.1.2.1

Move $a$ .

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

Step 7.1.2.2

Multiply $a$ by $a$ .

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

Step 7.1.3

Rewrite using the commutative property of multiplication.

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

Step 7.1.4

Multiply $5$ by $2$ .

$\frac{5 a^{2} + 10 a b - a (a - 2 b) - 4 a b}{(a + 2 b) (a - 2 b)}$

Step 7.1.5

Apply the distributive property.

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

Step 7.1.6

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

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Step 7.1.6.1

Move $a$ .

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

Step 7.1.6.2

Multiply $a$ by $a$ .

$\frac{5 a^{2} + 10 a b - a^{2} - a (- 2 b) - 4 a b}{(a + 2 b) (a - 2 b)}$

Step 7.1.7

Rewrite using the commutative property of multiplication.

$\frac{5 a^{2} + 10 a b - a^{2} - 1 \cdot - 2 a b - 4 a b}{(a + 2 b) (a - 2 b)}$

Step 7.1.8

Multiply $- 1$ by $- 2$ .

$\frac{5 a^{2} + 10 a b - a^{2} + 2 a b - 4 a b}{(a + 2 b) (a - 2 b)}$

Step 7.2

Subtract $a^{2}$ from $5 a^{2}$ .

$\frac{4 a^{2} + 10 a b + 2 a b - 4 a b}{(a + 2 b) (a - 2 b)}$

Step 7.3

Add $10 a b$ and $2 a b$ .

$\frac{4 a^{2} + 12 a b - 4 a b}{(a + 2 b) (a - 2 b)}$

Step 7.4

Subtract $4 a b$ from $12 a b$ .

$\frac{4 a^{2} + 8 a b}{(a + 2 b) (a - 2 b)}$

Step 8

Factor $4 a$ out of $4 a^{2} + 8 a b$ .

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Step 8.1

Factor $4 a$ out of $4 a^{2}$ .

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

Step 8.2

Factor $4 a$ out of $8 a b$ .

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

Step 8.3

Factor $4 a$ out of $4 a (a) + 4 a (2 b)$ .

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

Step 9

Cancel the common factor of $a + 2 b$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

$\frac{4 a}{a - 2 b}$