Algebra Examples

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Move $- 2$ to the left of $a$ .

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $4$ by $- 2$ .

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

Step 6.3

Add $- b$ and $4 b$ .

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

Step 7

To write $\frac{a b - 2 a + 3 b - 8}{b - 2}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 8

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

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

Step 9

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

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Step 9.1

Multiply $\frac{a b - 2 a + 3 b - 8}{b - 2}$ by $\frac{a}{a}$ .

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Simplify.

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Step 11.2.1

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

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Step 11.2.1.1

Move $a$ .

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

Step 11.2.1.2

Multiply $a$ by $a$ .

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

Step 11.2.2

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

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Step 11.2.2.1

Move $a$ .

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

Step 11.2.2.2

Multiply $a$ by $a$ .

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

Step 11.3

Apply the distributive property.

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

Step 11.4

Multiply $3$ by $2$ .

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

Step 11.5

Expand $(3 b + 6) (b - 2)$ using the FOIL Method.

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Step 11.5.1

Apply the distributive property.

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

Step 11.5.2

Apply the distributive property.

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b \cdot b + 3 b \cdot - 2 + 6 (b - 2)}{a (b - 2)}$

Step 11.5.3

Apply the distributive property.

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b \cdot b + 3 b \cdot - 2 + 6 b + 6 \cdot - 2}{a (b - 2)}$

Step 11.6

Simplify and combine like terms.

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Step 11.6.1

Simplify each term.

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Step 11.6.1.1

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

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Step 11.6.1.1.1

Move $b$ .

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 (b \cdot b) + 3 b \cdot - 2 + 6 b + 6 \cdot - 2}{a (b - 2)}$

Step 11.6.1.1.2

Multiply $b$ by $b$ .

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b^{2} + 3 b \cdot - 2 + 6 b + 6 \cdot - 2}{a (b - 2)}$

Step 11.6.1.2

Multiply $- 2$ by $3$ .

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b^{2} - 6 b + 6 b + 6 \cdot - 2}{a (b - 2)}$

Step 11.6.1.3

Multiply $6$ by $- 2$ .

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b^{2} - 6 b + 6 b - 12}{a (b - 2)}$

Step 11.6.2

Add $- 6 b$ and $6 b$ .

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b^{2} + 0 - 12}{a (b - 2)}$

Step 11.6.3

Add $3 b^{2}$ and $0$ .

$\frac{a^{2} b - 2 a^{2} + 3 b a - 8 a + 3 b^{2} - 12}{a (b - 2)}$