Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

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Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.1.3

Factor $a$ out of $a (2 a) + a (- 1 b)$ .

$\frac{b}{a (2 a - 1 b)} - \frac{4 a}{2 a b - b^{2}}$

Step 1.1.2

Rewrite $- 1 b$ as $- b$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $\frac{4 a}{b (2 a - b)}$ by $\frac{a}{a}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 6.5

Rewrite $4 a \cdot a$ as ${(2 a)}^{2}$ .

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

Step 6.6

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 $b = 2 a$ .

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

Step 6.7

Multiply $2$ by $- 1$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $b - 2 a$ and $2 a - b$ .

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Step 7.1.1

Factor $- 1$ out of $b$ .

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

Step 7.1.2

Factor $- 1$ out of $- 2 a$ .

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

Step 7.1.3

Factor $- 1$ out of $- 1 (- b) - (2 a)$ .

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

Step 7.1.4

Reorder terms.

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

Step 7.1.5

Cancel the common factor.

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

Step 7.1.6

Rewrite the expression.

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

Step 7.2

Simplify the expression.

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Step 7.2.1

Move $- 1$ to the left of $b + 2 a$ .

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

Step 7.2.2

Move the negative in front of the fraction.

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