Basic Math Examples

$\frac{a^{2} - 6 a + 9}{a^{2} - a - 12} \div \frac{a^{2} - a - 6}{a^{2} + 4 a + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - 6 a + 9}{a^{2} - a - 12} \cdot \frac{a^{2} + 4 a + 4}{a^{2} - a - 6}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 a = 2 \cdot a \cdot 3$

Step 2.3

Rewrite the polynomial.

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

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 3$ .

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

Step 3

Factor $a^{2} - a - 12$ 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 $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 3.2

Write the factored form using these integers.

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

Step 4

Factor using the perfect square rule.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 a = 2 \cdot a \cdot 2$

Step 4.3

Rewrite the polynomial.

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

Step 4.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 2$ .

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

Step 5

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

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Step 5.1

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 5.2

Write the factored form using these integers.

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

Step 6

Simplify terms.

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Step 6.1

Combine.

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

Step 6.2

Cancel the common factor of ${(a - 3)}^{2}$ and $a - 3$ .

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $a - 3$ out of $(a - 4) (a + 3) ((a - 3) (a + 2))$ .

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

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

Step 6.3

Cancel the common factor of ${(a + 2)}^{2}$ and $a + 2$ .

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Step 6.3.1

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

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

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $a + 2$ out of $(a - 4) (a + 3) (a + 2)$ .

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

Step 6.3.2.2

Cancel the common factor.

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

Step 6.3.2.3

Rewrite the expression.

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