Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

$\frac{3 a}{a^{2} - 3^{2}} - \frac{2 b}{a b + 3 b} - \frac{b}{a b - 3 b}$

Step 1.1.2

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 $b = 3$ .

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

Step 1.2

Factor $b$ out of $a b + 3 b$ .

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Step 1.2.1

Factor $b$ out of $a b$ .

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

Step 1.2.2

Factor $b$ out of $3 b$ .

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

Step 1.2.3

Factor $b$ out of $b a + b \cdot 3$ .

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

Step 1.3

Cancel the common factor of $b$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Factor $b$ out of $a b - 3 b$ .

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Step 1.4.1

Factor $b$ out of $a b$ .

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

Step 1.4.2

Factor $b$ out of $- 3 b$ .

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

Step 1.4.3

Factor $b$ out of $b a + b \cdot - 3$ .

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

Step 1.5

Cancel the common factor of $b$ .

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 2

Find the common denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $\frac{1}{a - 3}$ by $\frac{a + 3}{a + 3}$ .

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

Step 2.4

Multiply $\frac{1}{a - 3}$ by $\frac{a + 3}{a + 3}$ .

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

Step 2.5

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $- 2$ by $- 3$ .

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

Step 4.3

Apply the distributive property.

$\frac{3 a - 2 a + 6 - a - 1 \cdot 3}{(a + 3) (a - 3)}$

Step 4.4

Multiply $- 1$ by $3$ .

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

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $2 a$ from $3 a$ .

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

Step 5.2

Combine the opposite terms in $a + 6 - a - 3$ .

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Step 5.2.1

Subtract $a$ from $a$ .

$\frac{0 + 6 - 3}{(a + 3) (a - 3)}$

Step 5.2.2

Add $0$ and $6$ .

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

Step 5.3

Subtract $3$ from $6$ .

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