Algebra Examples

$\frac{4 a + 4}{3 a - 12} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \div \frac{4 a - 12}{3 a + 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 a + 4}{3 a - 12} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $4$ out of $4 a + 4$ .

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Step 2.1.1

Factor $4$ out of $4 a$ .

$\frac{4 (a) + 4}{3 a - 12} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 2.1.2

Factor $4$ out of $4$ .

$\frac{4 (a) + 4 (1)}{3 a - 12} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 2.1.3

Factor $4$ out of $4 (a) + 4 (1)$ .

$\frac{4 (a + 1)}{3 a - 12} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 2.2

Factor $3$ out of $3 a - 12$ .

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Step 2.2.1

Factor $3$ out of $3 a$ .

$\frac{4 (a + 1)}{3 (a) - 12} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 2.2.2

Factor $3$ out of $- 12$ .

$\frac{4 (a + 1)}{3 a + 3 \cdot - 4} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 2.2.3

Factor $3$ out of $3 a + 3 \cdot - 4$ .

$\frac{4 (a + 1)}{3 (a - 4)} \cdot \frac{a^{2} - 6 a + 8}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 3

Factor $a^{2} - 6 a + 8$ using the AC method.

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Step 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 3.2

Write the factored form using these integers.

$\frac{4 (a + 1)}{3 (a - 4)} \cdot \frac{(a - 4) (a - 2)}{a^{2} + 3 a - 4} \frac{3 a + 12}{4 a - 12}$

Step 4

Factor $a^{2} + 3 a - 4$ 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 4.2

Write the factored form using these integers.

$\frac{4 (a + 1)}{3 (a - 4)} \cdot \frac{(a - 4) (a - 2)}{(a - 1) (a + 4)} \frac{3 a + 12}{4 a - 12}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $a - 4$ .

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Step 5.1.1

Factor $a - 4$ out of $3 (a - 4)$ .

$\frac{4 (a + 1)}{(a - 4) \cdot 3} \cdot \frac{(a - 4) (a - 2)}{(a - 1) (a + 4)} \frac{3 a + 12}{4 a - 12}$

Step 5.1.2

Cancel the common factor.

$\frac{4 (a + 1)}{(a - 4) \cdot 3} \cdot \frac{(a - 4) (a - 2)}{(a - 1) (a + 4)} \frac{3 a + 12}{4 a - 12}$

Step 5.1.3

Rewrite the expression.

$\frac{4 (a + 1)}{3} \cdot \frac{a - 2}{(a - 1) (a + 4)} \frac{3 a + 12}{4 a - 12}$

Step 5.2

Multiply $\frac{4 (a + 1)}{3}$ by $\frac{a - 2}{(a - 1) (a + 4)}$ .

$\frac{4 (a + 1) (a - 2)}{3 ((a - 1) (a + 4))} \cdot \frac{3 a + 12}{4 a - 12}$

Step 5.3

Factor $3$ out of $3 a + 12$ .

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Step 5.3.1

Factor $3$ out of $3 a$ .

$\frac{4 (a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 (a) + 12}{4 a - 12}$

Step 5.3.2

Factor $3$ out of $12$ .

$\frac{4 (a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 a + 3 \cdot 4}{4 a - 12}$

Step 5.3.3

Factor $3$ out of $3 a + 3 \cdot 4$ .

$\frac{4 (a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{4 a - 12}$

Step 5.4

Factor $4$ out of $4 a - 12$ .

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Step 5.4.1

Factor $4$ out of $4 a$ .

$\frac{4 (a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{4 (a) - 12}$

Step 5.4.2

Factor $4$ out of $- 12$ .

$\frac{4 (a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{4 a + 4 \cdot - 3}$

Step 5.4.3

Factor $4$ out of $4 a + 4 \cdot - 3$ .

$\frac{4 (a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{4 (a - 3)}$

Step 5.5

Cancel the common factor of $4$ .

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Step 5.5.1

Factor $4$ out of $4 (a + 1) (a - 2)$ .

$\frac{4 ((a + 1) (a - 2))}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{4 (a - 3)}$

Step 5.5.2

Cancel the common factor.

$\frac{4 ((a + 1) (a - 2))}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{4 (a - 3)}$

Step 5.5.3

Rewrite the expression.

$\frac{(a + 1) (a - 2)}{3 (a - 1) (a + 4)} \cdot \frac{3 (a + 4)}{a - 3}$

Step 5.6

Cancel the common factor of $3 (a + 4)$ .

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Step 5.6.1

Factor $3 (a + 4)$ out of $3 (a - 1) (a + 4)$ .

$\frac{(a + 1) (a - 2)}{3 (a + 4) (a - 1)} \cdot \frac{3 (a + 4)}{a - 3}$

Step 5.6.2

Cancel the common factor.

$\frac{(a + 1) (a - 2)}{3 (a + 4) (a - 1)} \cdot \frac{3 (a + 4)}{a - 3}$

Step 5.6.3

Rewrite the expression.

$\frac{(a + 1) (a - 2)}{a - 1} \cdot \frac{1}{a - 3}$

Step 5.7

Multiply $\frac{(a + 1) (a - 2)}{a - 1}$ by $\frac{1}{a - 3}$ .

$\frac{(a + 1) (a - 2)}{(a - 1) (a - 3)}$