Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $b$ out of $a b$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

Factor $a$ out of $- a c$ .

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

Step 1.2.1.3

Factor $a$ out of $a \cdot a + a (- 1 c)$ .

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

Step 1.2.2

Rewrite $- 1 c$ as $- c$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Cancel the common factor of $a - b$ .

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Step 1.5.1

Move the leading negative in $- \frac{b (a - b)}{a (a - c)}$ into the numerator.

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

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

Cancel the common factor.

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

Step 1.5.5

Rewrite the expression.

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

Step 1.6

Cancel the common factor of $a - c$ .

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Step 1.6.1

Factor $a - c$ out of $a (a - c)$ .

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

Step 1.6.2

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

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

Step 1.6.3

Cancel the common factor.

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

Step 1.6.4

Rewrite the expression.

$\frac{b - c}{a + b} + \frac{- b}{a} \cdot \frac{a + c}{a + b}$

Step 1.7

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

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

Step 1.8

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Subtract $b a$ from $b a$ .

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

Step 5.4

Add $- c a$ and $0$ .

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

Step 5.5

Rewrite $- c a - b c$ in a factored form.

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Step 5.5.1

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

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Step 5.5.1.1

Factor $c$ out of $- c a$ .

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

Step 5.5.1.2

Factor $c$ out of $- b c$ .

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

Step 5.5.1.3

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

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

Step 5.5.2

Rewrite $- 1 a$ as $- a$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

Factor $- 1$ out of $- b$ .

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

Step 6.1.3

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

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

Step 6.1.4

Rewrite $- (a + b)$ as $- 1 (a + b)$ .

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

$\frac{c \cdot (- 1)}{a}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $- 1$ to the left of $c$ .

$\frac{- 1 \cdot c}{a}$

Step 6.2.2

Move the negative in front of the fraction.

$- \frac{c}{a}$