Algebra Examples

$\frac{c + 6 b}{a c + 2 b c - 6 a b - 3 a^{2}} + \frac{2 b}{a^{2} + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Reorder terms.

$\frac{c + 6 b}{- 3 a^{2} + a c + 2 b c - 6 a b} + \frac{2 b}{a^{2} + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{c + 6 b}{(- 3 a^{2} + a c) + 2 b c - 6 a b} + \frac{2 b}{a^{2} + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{c + 6 b}{a (- 3 a + c) + 2 b (- 3 a + c)} + \frac{2 b}{a^{2} + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $- 3 a + c$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a^{2} + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1.2

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

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Step 1.2.1

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

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a \cdot a + 2 a b} - \frac{b}{a c - 3 a^{2}}$

Step 1.2.2

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

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a \cdot a + a (2 b)} - \frac{b}{a c - 3 a^{2}}$

Step 1.2.3

Factor $a$ out of $a \cdot a + a (2 b)$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a (a + 2 b)} - \frac{b}{a c - 3 a^{2}}$

Step 1.3

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

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Step 1.3.1

Factor $a$ out of $a c$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a (a + 2 b)} - \frac{b}{a (c) - 3 a^{2}}$

Step 1.3.2

Factor $a$ out of $- 3 a^{2}$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a (a + 2 b)} - \frac{b}{a (c) + a (- 3 a)}$

Step 1.3.3

Factor $a$ out of $a (c) + a (- 3 a)$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} + \frac{2 b}{a (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 2

To write $\frac{c + 6 b}{(- 3 a + c) (a + 2 b)}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} \cdot \frac{a}{a} + \frac{2 b}{a (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 3

To write $\frac{2 b}{a (a + 2 b)}$ as a fraction with a common denominator, multiply by $\frac{- 3 a + c}{- 3 a + c}$ .

$\frac{c + 6 b}{(- 3 a + c) (a + 2 b)} \cdot \frac{a}{a} + \frac{2 b}{a (a + 2 b)} \cdot \frac{- 3 a + c}{- 3 a + c} - \frac{b}{a (c - 3 a)}$

Step 4

Write each expression with a common denominator of $(- 3 a + c) (a + 2 b) a$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{c + 6 b}{(- 3 a + c) (a + 2 b)}$ by $\frac{a}{a}$ .

$\frac{(c + 6 b) a}{(- 3 a + c) (a + 2 b) a} + \frac{2 b}{a (a + 2 b)} \cdot \frac{- 3 a + c}{- 3 a + c} - \frac{b}{a (c - 3 a)}$

Step 4.2

Multiply $\frac{2 b}{a (a + 2 b)}$ by $\frac{- 3 a + c}{- 3 a + c}$ .

$\frac{(c + 6 b) a}{(- 3 a + c) (a + 2 b) a} + \frac{2 b (- 3 a + c)}{a (a + 2 b) (- 3 a + c)} - \frac{b}{a (c - 3 a)}$

Step 4.3

Reorder the factors of $(- 3 a + c) (a + 2 b) a$ .

$\frac{(c + 6 b) a}{a (- 3 a + c) (a + 2 b)} + \frac{2 b (- 3 a + c)}{a (a + 2 b) (- 3 a + c)} - \frac{b}{a (c - 3 a)}$

Step 4.4

Reorder the factors of $a (a + 2 b) (- 3 a + c)$ .

$\frac{(c + 6 b) a}{a (- 3 a + c) (a + 2 b)} + \frac{2 b (- 3 a + c)}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(c + 6 b) a + 2 b (- 3 a + c)}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$\frac{c a + 6 b a + 2 b (- 3 a + c)}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 6.1.2

Apply the distributive property.

$\frac{c a + 6 b a + 2 b (- 3 a) + 2 b c}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 6.1.3

Rewrite using the commutative property of multiplication.

$\frac{c a + 6 b a + 2 \cdot - 3 b a + 2 b c}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 6.1.4

Multiply $2$ by $- 3$ .

$\frac{c a + 6 b a - 6 b a + 2 b c}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 6.1.5

Subtract $6 b a$ from $6 b a$ .

$\frac{c a + 0 + 2 b c}{a (- 3 a + c) (a + 2 b)} - \frac{b}{a (c - 3 a)}$

Step 6.1.6

Add $c a$ and $0$ .

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

Step 6.1.7

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

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Step 6.1.7.1

Factor $c$ out of $c a$ .

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

Step 6.1.7.2

Factor $c$ out of $2 b c$ .

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

Step 6.1.7.3

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

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

Step 6.2

Cancel the common factor of $a + 2 b$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

Simplify terms.

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Step 7.1

Reorder terms.

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 7.3

Factor $- 1$ out of $- 3 a$ .

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

Step 7.4

Factor $- 1$ out of $c$ .

$\frac{c - b}{a (- (3 a) - 1 (- c))}$

Step 7.5

Factor $- 1$ out of $- (3 a) - 1 (- c)$ .

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

Step 7.6

Rewrite negatives.

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Step 7.6.1

Rewrite $- (3 a - c)$ as $- 1 (3 a - c)$ .

$\frac{c - b}{a (- 1 (3 a - c))}$

Step 7.6.2

Move the negative in front of the fraction.

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