Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Factor $- 1$ out of $b$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Reorder terms.

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

Step 6

To write $\frac{b}{{(a - b)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{b \cdot - 1}{b \cdot - 1}$ .

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

Step 7

To write $- \frac{a + b}{b (- 1 (a - b))}$ as a fraction with a common denominator, multiply by $\frac{a - b}{a - b}$ .

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

Step 8

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

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Step 8.1

Multiply $\frac{b}{{(a - b)}^{2}}$ by $\frac{b \cdot - 1}{b \cdot - 1}$ .

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

Step 8.2

Multiply $\frac{a + b}{b (- 1 (a - b))}$ by $\frac{a - b}{a - b}$ .

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

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

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

Step 8.6

Add $1$ and $1$ .

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

Step 8.7

Reorder the factors of ${(a - b)}^{2} (b \cdot - 1)$ .

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

Step 8.8

Reorder the factors of $b (- 1 {(a - b)}^{2})$ .

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Rewrite using the commutative property of multiplication.

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

Step 10.2

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

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Step 10.2.1

Move $b$ .

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

Step 10.2.2

Multiply $b$ by $b$ .

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

Step 10.3

Apply the distributive property.

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

Step 10.4

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

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Step 10.4.1

Apply the distributive property.

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

Step 10.4.2

Apply the distributive property.

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

Step 10.4.3

Apply the distributive property.

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

Step 10.5

Simplify and combine like terms.

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Step 10.5.1

Simplify each term.

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Step 10.5.1.1

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

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Step 10.5.1.1.1

Move $a$ .

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

Step 10.5.1.1.2

Multiply $a$ by $a$ .

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

Step 10.5.1.2

Rewrite using the commutative property of multiplication.

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

Step 10.5.1.3

Multiply $- 1$ by $- 1$ .

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

Step 10.5.1.4

Multiply $a$ by $1$ .

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

Step 10.5.1.5

Rewrite using the commutative property of multiplication.

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

Step 10.5.1.6

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

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Step 10.5.1.6.1

Move $b$ .

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

Step 10.5.1.6.2

Multiply $b$ by $b$ .

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

Step 10.5.1.7

Multiply $- 1$ by $- 1$ .

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

Step 10.5.1.8

Multiply $b^{2}$ by $1$ .

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

Step 10.5.2

Subtract $b a$ from $a b$ .

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Step 10.5.2.1

Move $b$ .

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

Step 10.5.2.2

Subtract $a b$ from $a b$ .

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

Step 10.5.3

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

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

Step 10.6

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

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

Step 10.7

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

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

Step 11

Dividing two negative values results in a positive value.

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