Algebra Examples

$\frac{6}{m^{2} + 3 m} - \frac{1}{m^{2} - m} + \frac{3}{m^{2} + 2 m - 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $m$ out of $m^{2} + 3 m$ .

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Step 1.1.1

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

$\frac{6}{m \cdot m + 3 m} - \frac{1}{m^{2} - m} + \frac{3}{m^{2} + 2 m - 3}$

Step 1.1.2

Factor $m$ out of $3 m$ .

$\frac{6}{m \cdot m + m \cdot 3} - \frac{1}{m^{2} - m} + \frac{3}{m^{2} + 2 m - 3}$

Step 1.1.3

Factor $m$ out of $m \cdot m + m \cdot 3$ .

$\frac{6}{m (m + 3)} - \frac{1}{m^{2} - m} + \frac{3}{m^{2} + 2 m - 3}$

Step 1.2

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

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Step 1.2.1

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

$\frac{6}{m (m + 3)} - \frac{1}{m \cdot m - m} + \frac{3}{m^{2} + 2 m - 3}$

Step 1.2.2

Factor $m$ out of $- m$ .

$\frac{6}{m (m + 3)} - \frac{1}{m \cdot m + m \cdot - 1} + \frac{3}{m^{2} + 2 m - 3}$

Step 1.2.3

Factor $m$ out of $m \cdot m + m \cdot - 1$ .

$\frac{6}{m (m + 3)} - \frac{1}{m (m - 1)} + \frac{3}{m^{2} + 2 m - 3}$

Step 1.3

Factor $m^{2} + 2 m - 3$ using the AC method.

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Step 1.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.3.2

Write the factored form using these integers.

$\frac{6}{m (m + 3)} - \frac{1}{m (m - 1)} + \frac{3}{(m - 1) (m + 3)}$

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{6}{m (m + 3)}$ by $\frac{m - 1}{m - 1}$ .

$\frac{6}{m (m + 3)} \cdot \frac{m - 1}{m - 1} - \frac{1}{m (m - 1)} + \frac{3}{(m - 1) (m + 3)}$

Step 2.2

Multiply $\frac{6}{m (m + 3)}$ by $\frac{m - 1}{m - 1}$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - \frac{1}{m (m - 1)} + \frac{3}{(m - 1) (m + 3)}$

Step 2.3

Multiply $\frac{1}{m (m - 1)}$ by $\frac{m + 3}{m + 3}$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - (\frac{1}{m (m - 1)} \cdot \frac{m + 3}{m + 3}) + \frac{3}{(m - 1) (m + 3)}$

Step 2.4

Multiply $\frac{1}{m (m - 1)}$ by $\frac{m + 3}{m + 3}$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - \frac{m + 3}{m (m - 1) (m + 3)} + \frac{3}{(m - 1) (m + 3)}$

Step 2.5

Multiply $\frac{3}{(m - 1) (m + 3)}$ by $\frac{m}{m}$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - \frac{m + 3}{m (m - 1) (m + 3)} + \frac{3}{(m - 1) (m + 3)} \cdot \frac{m}{m}$

Step 2.6

Multiply $\frac{3}{(m - 1) (m + 3)}$ by $\frac{m}{m}$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - \frac{m + 3}{m (m - 1) (m + 3)} + \frac{3 m}{(m - 1) (m + 3) m}$

Step 2.7

Reorder the factors of $m (m - 1) (m + 3)$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - \frac{m + 3}{m (m + 3) (m - 1)} + \frac{3 m}{(m - 1) (m + 3) m}$

Step 2.8

Reorder the factors of $(m - 1) (m + 3) m$ .

$\frac{6 (m - 1)}{m (m + 3) (m - 1)} - \frac{m + 3}{m (m + 3) (m - 1)} + \frac{3 m}{m (m + 3) (m - 1)}$

Step 3

Combine the numerators over the common denominator.

$\frac{6 (m - 1) - (m + 3) + 3 m}{m (m + 3) (m - 1)}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{6 m + 6 \cdot - 1 - (m + 3) + 3 m}{m (m + 3) (m - 1)}$

Step 4.2

Multiply $6$ by $- 1$ .

$\frac{6 m - 6 - (m + 3) + 3 m}{m (m + 3) (m - 1)}$

Step 4.3

Apply the distributive property.

$\frac{6 m - 6 - m - 1 \cdot 3 + 3 m}{m (m + 3) (m - 1)}$

Step 4.4

Multiply $- 1$ by $3$ .

$\frac{6 m - 6 - m - 3 + 3 m}{m (m + 3) (m - 1)}$

Step 5

Simplify by adding terms.

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Step 5.1

Subtract $m$ from $6 m$ .

$\frac{5 m - 6 - 3 + 3 m}{m (m + 3) (m - 1)}$

Step 5.2

Add $5 m$ and $3 m$ .

$\frac{8 m - 6 - 3}{m (m + 3) (m - 1)}$

Step 5.3

Subtract $3$ from $- 6$ .

$\frac{8 m - 9}{m (m + 3) (m - 1)}$