Algebra Examples

$\frac{c}{b - c} + \frac{b^{2} - 3 b c}{b^{2} - c^{2}}$

Step 1

Simplify each term.

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Step 1.1

Factor $b$ out of $b^{2} - 3 b c$ .

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Step 1.1.1

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

$\frac{c}{b - c} + \frac{b \cdot b - 3 b c}{b^{2} - c^{2}}$

Step 1.1.2

Factor $b$ out of $- 3 b c$ .

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

Step 1.1.3

Factor $b$ out of $b \cdot b + b (- 3 c)$ .

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

Step 1.2

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 $b = c$ .

$\frac{c}{b - c} + \frac{b (b - 3 c)}{(b + c) (b - c)}$

Step 2

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

$\frac{c}{b - c} \cdot \frac{b + c}{b + c} + \frac{b (b - 3 c)}{(b + c) (b - c)}$

Step 3

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

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Step 3.1

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

$\frac{c (b + c)}{(b - c) (b + c)} + \frac{b (b - 3 c)}{(b + c) (b - c)}$

Step 3.2

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

$\frac{c (b + c)}{(b + c) (b - c)} + \frac{b (b - 3 c)}{(b + c) (b - c)}$

Step 4

Combine the numerators over the common denominator.

$\frac{c (b + c) + b (b - 3 c)}{(b + c) (b - c)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{c b + c \cdot c + b (b - 3 c)}{(b + c) (b - c)}$

Step 5.2

Multiply $c$ by $c$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $b$ by $b$ .

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

Step 5.5

Rewrite using the commutative property of multiplication.

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

Step 5.6

Subtract $3 b c$ from $c b$ .

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Step 5.6.1

Reorder $c$ and $b$ .

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

Step 5.6.2

Subtract $3 b c$ from $b c$ .

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

Step 5.7

Factor using the perfect square rule.

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Step 5.7.1

Rearrange terms.

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

Step 5.7.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 b c = 2 \cdot c \cdot b$

Step 5.7.3

Rewrite the polynomial.

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

Step 5.7.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = c$ and $b = b$ .

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

Step 6

Cancel the common factor of ${(c - b)}^{2}$ and $b - c$ .

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Step 6.1

Factor $- 1$ out of $c$ .

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Apply the product rule to $- 1 (- c + b)$ .

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

Step 6.5

Raise $- 1$ to the power of $2$ .

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

Step 6.6

Multiply ${(- c + b)}^{2}$ by $1$ .

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

Step 6.7

Reorder terms.

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

Step 6.8

Factor $b - c$ out of ${(b - c)}^{2}$ .

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

Step 6.9

Cancel the common factors.

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Step 6.9.1

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

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

Step 6.9.2

Cancel the common factor.

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

Step 6.9.3

Rewrite the expression.

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