Pre-Algebra Examples

$\frac{a^{2} - a - 2}{a^{2} - a - 6} \div \frac{a^{2} - 2 a}{2 a + a^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

$- 2, 1$

Step 2.2

Write the factored form using these integers.

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

Step 3

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

$- 3, 2$

Step 3.2

Write the factored form using these integers.

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

Step 4

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

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Step 4.1

Factor $a$ out of $2 a$ .

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

Step 4.2

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

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

Step 4.3

Factor $a$ out of $a \cdot 2 + a \cdot a$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Factor $a$ out of $- 2 a$ .

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

Step 5.3

Factor $a$ out of $a \cdot a + a \cdot - 2$ .

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

Step 6

Cancel the common factor of $a - 2$ .

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Step 6.1

Factor $a - 2$ out of $a (a - 2)$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

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

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

Step 8

Cancel the common factor of $a$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of $2 + a$ and $a + 2$ .

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Step 9.1

Reorder terms.

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

$\frac{a + 1}{a - 3}$