Basic Math Examples

$\frac{a^{2} + 3 a}{3 a - 9} \cdot \frac{a^{2} - a - 6}{2 a^{2} + 6 a}$

Step 1

Simplify with factoring out.

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Step 1.1

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

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Step 1.1.1

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

$\frac{a \cdot a + 3 a}{3 a - 9} \cdot \frac{a^{2} - a - 6}{2 a^{2} + 6 a}$

Step 1.1.2

Factor $a$ out of $3 a$ .

$\frac{a \cdot a + a \cdot 3}{3 a - 9} \cdot \frac{a^{2} - a - 6}{2 a^{2} + 6 a}$

Step 1.1.3

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

$\frac{a (a + 3)}{3 a - 9} \cdot \frac{a^{2} - a - 6}{2 a^{2} + 6 a}$

Step 1.2

Factor $3$ out of $3 a - 9$ .

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Step 1.2.1

Factor $3$ out of $3 a$ .

$\frac{a (a + 3)}{3 (a) - 9} \cdot \frac{a^{2} - a - 6}{2 a^{2} + 6 a}$

Step 1.2.2

Factor $3$ out of $- 9$ .

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

Step 1.2.3

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

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

Step 2

Factor $a^{2} - a - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

Factor $2 a$ out of $2 a^{2} + 6 a$ .

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Step 3.1.1

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

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

Step 3.1.2

Factor $2 a$ out of $6 a$ .

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

Step 3.1.3

Factor $2 a$ out of $2 a (a) + 2 a (3)$ .

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

Step 3.2

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

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Step 3.2.1

Factor $a (a + 3)$ out of $2 a (a + 3)$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $a - 3$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{a + 2}{2}$

Step 3.4

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

$\frac{a + 2}{3 \cdot 2}$

Step 3.5

Multiply $3$ by $2$ .

$\frac{a + 2}{6}$