Algebra Examples

$\frac{m^{2} - 2 m - 8}{8 m + 24} \div \frac{2 m - 8}{m^{2} + 7 m + 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{m^{2} - 2 m - 8}{8 m + 24} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Step 2

Factor $m^{2} - 2 m - 8$ using the AC method.

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Step 2.1

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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(m - 4) (m + 2)}{8 m + 24} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Step 3

Factor $8$ out of $8 m + 24$ .

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Step 3.1

Factor $8$ out of $8 m$ .

$\frac{(m - 4) (m + 2)}{8 (m) + 24} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Step 3.2

Factor $8$ out of $24$ .

$\frac{(m - 4) (m + 2)}{8 m + 8 \cdot 3} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Step 3.3

Factor $8$ out of $8 m + 8 \cdot 3$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{m^{2} + 7 m + 12}{2 m - 8}$

Step 4

Factor $m^{2} + 7 m + 12$ using the AC method.

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Step 4.1

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 $12$ and whose sum is $7$ .

$3, 4$

Step 4.2

Write the factored form using these integers.

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 m - 8}$

Step 5

Factor $2$ out of $2 m - 8$ .

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Step 5.1

Factor $2$ out of $2 m$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 (m) - 8}$

Step 5.2

Factor $2$ out of $- 8$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 m + 2 \cdot - 4}$

Step 5.3

Factor $2$ out of $2 m + 2 \cdot - 4$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2 (m - 4)}$

Step 6

Cancel the common factor of $m - 4$ .

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Step 6.1

Factor $m - 4$ out of $2 (m - 4)$ .

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{(m - 4) \cdot 2}$

Step 6.2

Cancel the common factor.

$\frac{(m - 4) (m + 2)}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{(m - 4) \cdot 2}$

Step 6.3

Rewrite the expression.

$\frac{m + 2}{8 (m + 3)} \cdot \frac{(m + 3) (m + 4)}{2}$

Step 7

Cancel the common factor of $m + 3$ .

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Step 7.1

Factor $m + 3$ out of $8 (m + 3)$ .

$\frac{m + 2}{(m + 3) \cdot 8} \cdot \frac{(m + 3) (m + 4)}{2}$

Step 7.2

Cancel the common factor.

$\frac{m + 2}{(m + 3) \cdot 8} \cdot \frac{(m + 3) (m + 4)}{2}$

Step 7.3

Rewrite the expression.

$\frac{m + 2}{8} \cdot \frac{m + 4}{2}$

Step 8

Multiply $\frac{m + 2}{8}$ by $\frac{m + 4}{2}$ .

$\frac{(m + 2) (m + 4)}{8 \cdot 2}$

Step 9

Multiply $8$ by $2$ .

$\frac{(m + 2) (m + 4)}{16}$

Step 10

Expand $(m + 2) (m + 4)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$\frac{m (m + 4) + 2 (m + 4)}{16}$

Step 10.2

Apply the distributive property.

$\frac{m \cdot m + m \cdot 4 + 2 (m + 4)}{16}$

Step 10.3

Apply the distributive property.

$\frac{m \cdot m + m \cdot 4 + 2 m + 2 \cdot 4}{16}$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $m$ by $m$ .

$\frac{m^{2} + m \cdot 4 + 2 m + 2 \cdot 4}{16}$

Step 11.1.2

Move $4$ to the left of $m$ .

$\frac{m^{2} + 4 \cdot m + 2 m + 2 \cdot 4}{16}$

Step 11.1.3

Multiply $2$ by $4$ .

$\frac{m^{2} + 4 m + 2 m + 8}{16}$

Step 11.2

Add $4 m$ and $2 m$ .

$\frac{m^{2} + 6 m + 8}{16}$

Step 12

Split the fraction $\frac{m^{2} + 6 m + 8}{16}$ into two fractions.

$\frac{m^{2} + 6 m}{16} + \frac{8}{16}$

Step 13

Split the fraction $\frac{m^{2} + 6 m}{16}$ into two fractions.

$\frac{m^{2}}{16} + \frac{6 m}{16} + \frac{8}{16}$

Step 14

Cancel the common factor of $6$ and $16$ .

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Step 14.1

Factor $2$ out of $6 m$ .

$\frac{m^{2}}{16} + \frac{2 (3 m)}{16} + \frac{8}{16}$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $2$ out of $16$ .

$\frac{m^{2}}{16} + \frac{2 (3 m)}{2 (8)} + \frac{8}{16}$

Step 14.2.2

Cancel the common factor.

$\frac{m^{2}}{16} + \frac{2 (3 m)}{2 \cdot 8} + \frac{8}{16}$

Step 14.2.3

Rewrite the expression.

$\frac{m^{2}}{16} + \frac{3 m}{8} + \frac{8}{16}$

Step 15

Cancel the common factor of $8$ and $16$ .

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Step 15.1

Factor $8$ out of $8$ .

$\frac{m^{2}}{16} + \frac{3 m}{8} + \frac{8 (1)}{16}$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $8$ out of $16$ .

$\frac{m^{2}}{16} + \frac{3 m}{8} + \frac{8 \cdot 1}{8 \cdot 2}$

Step 15.2.2

Cancel the common factor.

$\frac{m^{2}}{16} + \frac{3 m}{8} + \frac{8 \cdot 1}{8 \cdot 2}$

Step 15.2.3

Rewrite the expression.

$\frac{m^{2}}{16} + \frac{3 m}{8} + \frac{1}{2}$