Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Reorder the factors of $b a$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $b$ by $b$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $2$ by $- 1$ .

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

Step 5.5

Apply the distributive property.

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

Step 5.6

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

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Step 5.6.1

Move $a$ .

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

Step 5.6.2

Multiply $a$ by $a$ .

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

Step 5.7

Subtract $2 b a$ from $2 a b$ .

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Step 5.7.1

Move $b$ .

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

Step 5.7.2

Subtract $2 a b$ from $2 a b$ .

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

Step 5.8

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

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

Step 5.9

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

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

Step 6

Simplify each term.

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Step 6.1

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

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Step 6.1.1

Factor $b$ out of $b^{2}$ .

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

Step 6.1.2

Factor $b$ out of $a b$ .

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

Step 6.1.3

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

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

Step 6.2

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

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Step 6.2.1

Factor $b$ out of $b^{2}$ .

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

Step 6.2.2

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

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

Step 6.2.3

Factor $b$ out of $b \cdot b + b (- a)$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

Reorder terms.

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

Step 11.2

Raise $a + b$ to the power of $1$ .

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

Step 11.3

Raise $a + b$ to the power of $1$ .

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

Step 11.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 11.5

Add $1$ and $1$ .

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

Step 11.6

Rewrite $(- a + b) (b - a)$ as ${(- a + b)}^{2}$ .

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

Step 11.7

Reorder $- {(- a + b)}^{2}$ and ${(a + b)}^{2}$ .

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

Step 11.8

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

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

Step 11.9

Simplify.

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Step 11.9.1

Subtract $a$ from $a$ .

$\frac{(b + a) (b - a)}{a b} \cdot \frac{(b + 0 + b) (a + b - (- a + b))}{b (b + a) (b - a)}$

Step 11.9.2

Add $b$ and $0$ .

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

Step 11.9.3

Add $b$ and $b$ .

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

Step 11.9.4

Apply the distributive property.

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

Step 11.9.5

Multiply $- - a$ .

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Step 11.9.5.1

Multiply $- 1$ by $- 1$ .

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

Step 11.9.5.2

Multiply $a$ by $1$ .

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

Step 11.9.6

Add $a$ and $a$ .

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

Step 11.9.7

Subtract $b$ from $b$ .

$\frac{(b + a) (b - a)}{a b} \cdot \frac{2 b (2 a + 0)}{b (b + a) (b - a)}$

Step 11.9.8

Add $2 a$ and $0$ .

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

Step 11.9.9

Multiply $2$ by $2$ .

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

Step 12

Reduce the expression by cancelling the common factors.

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Step 12.1

Cancel the common factor of $(b + a) (b - a)$ .

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Step 12.1.1

Factor $(b + a) (b - a)$ out of $b (b + a) (b - a)$ .

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

Step 12.1.2

Cancel the common factor.

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

Step 12.1.3

Rewrite the expression.

$\frac{1}{a b} \cdot \frac{4 b a}{b}$

Step 12.2

Cancel the common factor of $b a$ .

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Step 12.2.1

Factor $b a$ out of $a b$ .

$\frac{1}{b a (1)} \cdot \frac{4 b a}{b}$

Step 12.2.2

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

$\frac{1}{b a (1)} \cdot \frac{b a (4)}{b}$

Step 12.2.3

Cancel the common factor.

$\frac{1}{b a \cdot 1} \cdot \frac{b a \cdot 4}{b}$

Step 12.2.4

Rewrite the expression.

$\frac{4}{b}$