Algebra Examples

$\frac{3 c + 6}{3 c + 9} \div \frac{c^{2} - 6 c + 8}{c^{2} + 2 c - 3} \cdot \frac{3 c - 6}{3 c + 12}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $3$ out of $3 c + 6$ .

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Step 1.1.1

Factor $3$ out of $3 c$ .

$\frac{3 (c) + 6}{3 c + 9} \div \frac{c^{2} - 6 c + 8}{c^{2} + 2 c - 3} \cdot \frac{3 c - 6}{3 c + 12}$

Step 1.1.2

Factor $3$ out of $6$ .

$\frac{3 c + 3 \cdot 2}{3 c + 9} \div \frac{c^{2} - 6 c + 8}{c^{2} + 2 c - 3} \cdot \frac{3 c - 6}{3 c + 12}$

Step 1.1.3

Factor $3$ out of $3 c + 3 \cdot 2$ .

$\frac{3 (c + 2)}{3 c + 9} \div \frac{c^{2} - 6 c + 8}{c^{2} + 2 c - 3} \cdot \frac{3 c - 6}{3 c + 12}$

Step 1.2

Factor $3$ out of $3 c + 9$ .

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Step 1.2.1

Factor $3$ out of $3 c$ .

$\frac{3 (c + 2)}{3 (c) + 9} \div \frac{c^{2} - 6 c + 8}{c^{2} + 2 c - 3} \cdot \frac{3 c - 6}{3 c + 12}$

Step 1.2.2

Factor $3$ out of $9$ .

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

Step 1.2.3

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

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

Step 2

Factor $c^{2} - 6 c + 8$ 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 $- 6$ .

$- 4, - 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

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

$- 1, 3$

Step 3.2

Write the factored form using these integers.

$\frac{3 (c + 2)}{3 (c + 3)} \div \frac{(c - 4) (c - 2)}{(c - 1) (c + 3)} \cdot \frac{3 c - 6}{3 c + 12}$

Step 4

Simplify terms.

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Step 4.1

Reduce the expression $\frac{3 (c + 2)}{3 (c + 3)}$ by cancelling the common factors.

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Step 4.1.1

Cancel the common factor.

$\frac{3 (c + 2)}{3 (c + 3)} \div \frac{(c - 4) (c - 2)}{(c - 1) (c + 3)} \cdot \frac{3 c - 6}{3 c + 12}$

Step 4.1.2

Rewrite the expression.

$\frac{c + 2}{c + 3} \div \frac{(c - 4) (c - 2)}{(c - 1) (c + 3)} \cdot \frac{3 c - 6}{3 c + 12}$

Step 4.2

Factor $3$ out of $3 c - 6$ .

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Step 4.2.1

Factor $3$ out of $3 c$ .

$\frac{c + 2}{c + 3} \div \frac{(c - 4) (c - 2)}{(c - 1) (c + 3)} \cdot \frac{3 (c) - 6}{3 c + 12}$

Step 4.2.2

Factor $3$ out of $- 6$ .

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

Step 4.2.3

Factor $3$ out of $3 c + 3 \cdot - 2$ .

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

Step 4.3

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

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Step 4.3.1

Factor $3$ out of $3 c$ .

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

Step 4.3.2

Factor $3$ out of $12$ .

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

Step 4.3.3

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

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

Step 5

To divide by a fraction, multiply by its reciprocal.

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

Step 6

Cancel the common factor of $c + 3$ .

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Cancel the common factor of $c - 2$ .

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Step 8.1

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

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

Step 8.2

Factor $c - 2$ out of $3 (c - 2)$ .

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

Step 8.3

Cancel the common factor.

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

Step 8.4

Rewrite the expression.

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

Step 9

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

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

Step 10

Cancel the common factor of $3$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

$\frac{(c + 2) (c - 1)}{(c - 4) (c + 4)}$