Basic Math Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(3 a + 2 b) (2 a - b)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.2

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

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Step 5.2.1.2.1

Move $a$ .

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

Step 5.2.1.2.2

Multiply $a$ by $a$ .

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

Step 5.2.1.3

Multiply $3$ by $2$ .

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

Step 5.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.5

Multiply $3$ by $- 1$ .

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

Step 5.2.1.6

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.7

Multiply $2$ by $2$ .

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

Step 5.2.1.8

Rewrite using the commutative property of multiplication.

$\frac{6 a^{2} - 3 a b + 4 b a + 2 \cdot - 1 b \cdot b - (3 a - 2 b) (2 + b)}{(2 + b) (2 a - b)}$

Step 5.2.1.9

Multiply $b$ by $b$ by adding the exponents.

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Step 5.2.1.9.1

Move $b$ .

$\frac{6 a^{2} - 3 a b + 4 b a + 2 \cdot - 1 (b \cdot b) - (3 a - 2 b) (2 + b)}{(2 + b) (2 a - b)}$

Step 5.2.1.9.2

Multiply $b$ by $b$ .

$\frac{6 a^{2} - 3 a b + 4 b a + 2 \cdot - 1 b^{2} - (3 a - 2 b) (2 + b)}{(2 + b) (2 a - b)}$

Step 5.2.1.10

Multiply $2$ by $- 1$ .

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

Step 5.2.2

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

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Step 5.2.2.1

Move $b$ .

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

Step 5.2.2.2

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

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

Step 5.3

Multiply $a$ by $1$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $3$ by $- 1$ .

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

Step 5.6

Multiply $- 2$ by $- 1$ .

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

Step 5.7

Expand $(- 3 a + 2 b) (2 + b)$ using the FOIL Method.

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Step 5.7.1

Apply the distributive property.

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

Step 5.7.2

Apply the distributive property.

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

Step 5.7.3

Apply the distributive property.

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

Step 5.8

Simplify each term.

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Step 5.8.1

Multiply $2$ by $- 3$ .

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

Step 5.8.2

Multiply $2$ by $2$ .

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

Step 5.8.3

Multiply $b$ by $b$ by adding the exponents.

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Step 5.8.3.1

Move $b$ .

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

Step 5.8.3.2

Multiply $b$ by $b$ .

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

Step 5.9

Subtract $3 a b$ from $a b$ .

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

Step 5.10

Add $- 2 b^{2}$ and $2 b^{2}$ .

$\frac{6 a^{2} - 6 a - 2 a b + 4 b + 0}{(2 + b) (2 a - b)}$

Step 5.11

Add $6 a^{2} - 6 a - 2 a b + 4 b$ and $0$ .

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

Step 5.12

Factor $2$ out of $6 a^{2} - 6 a - 2 a b + 4 b$ .

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Step 5.12.1

Factor $2$ out of $6 a^{2}$ .

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

Step 5.12.2

Factor $2$ out of $- 6 a$ .

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

Step 5.12.3

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

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

Step 5.12.4

Factor $2$ out of $4 b$ .

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

Step 5.12.5

Factor $2$ out of $2 (3 a^{2}) + 2 (- 3 a)$ .

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

Step 5.12.6

Factor $2$ out of $2 (3 a^{2} - 3 a) + 2 (- a b)$ .

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

Step 5.12.7

Factor $2$ out of $2 (3 a^{2} - 3 a - a b) + 2 (2 b)$ .

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