Basic Math Examples

$\frac{y^{2} - 4 y - 12}{2 y^{2} - 72} \div \frac{y^{2} + 8 y + 12}{y^{2} + 12 y + 36}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{y^{2} - 4 y - 12}{2 y^{2} - 72} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 2

Factor $y^{2} - 4 y - 12$ 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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(y - 6) (y + 2)}{2 y^{2} - 72} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $2$ out of $2 y^{2} - 72$ .

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Step 3.1.1

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

$\frac{(y - 6) (y + 2)}{2 (y^{2}) - 72} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 3.1.2

Factor $2$ out of $- 72$ .

$\frac{(y - 6) (y + 2)}{2 y^{2} + 2 \cdot - 36} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 3.1.3

Factor $2$ out of $2 y^{2} + 2 \cdot - 36$ .

$\frac{(y - 6) (y + 2)}{2 (y^{2} - 36)} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 3.2

Rewrite $36$ as $6^{2}$ .

$\frac{(y - 6) (y + 2)}{2 (y^{2} - 6^{2})} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 3.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 6$ .

$\frac{(y - 6) (y + 2)}{2 (y + 6) (y - 6)} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 4

Cancel the common factor of $y - 6$ .

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Step 4.1

Cancel the common factor.

$\frac{(y - 6) (y + 2)}{2 (y + 6) (y - 6)} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 4.2

Rewrite the expression.

$\frac{y + 2}{2 (y + 6)} \cdot \frac{y^{2} + 12 y + 36}{y^{2} + 8 y + 12}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $36$ as $6^{2}$ .

$\frac{y + 2}{2 (y + 6)} \cdot \frac{y^{2} + 12 y + 6^{2}}{y^{2} + 8 y + 12}$

Step 5.2

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

$12 y = 2 \cdot y \cdot 6$

Step 5.3

Rewrite the polynomial.

$\frac{y + 2}{2 (y + 6)} \cdot \frac{y^{2} + 2 \cdot y \cdot 6 + 6^{2}}{y^{2} + 8 y + 12}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 6$ .

$\frac{y + 2}{2 (y + 6)} \cdot \frac{{(y + 6)}^{2}}{y^{2} + 8 y + 12}$

Step 6

Factor $y^{2} + 8 y + 12$ using the AC method.

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Step 6.1

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 $8$ .

$2, 6$

Step 6.2

Write the factored form using these integers.

$\frac{y + 2}{2 (y + 6)} \cdot \frac{{(y + 6)}^{2}}{(y + 2) (y + 6)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $y + 2$ .

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Step 7.1.1

Cancel the common factor.

$\frac{y + 2}{2 (y + 6)} \cdot \frac{{(y + 6)}^{2}}{(y + 2) (y + 6)}$

Step 7.1.2

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $y + 6$ .

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Step 7.2.1

Factor $y + 6$ out of $2 (y + 6)$ .

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

Step 7.2.2

Factor $y + 6$ out of ${(y + 6)}^{2}$ .

$\frac{1}{(y + 6) \cdot 2} \cdot \frac{(y + 6) (y + 6)}{y + 6}$

Step 7.2.3

Cancel the common factor.

$\frac{1}{(y + 6) \cdot 2} \cdot \frac{(y + 6) (y + 6)}{y + 6}$

Step 7.2.4

Rewrite the expression.

$\frac{1}{2} \cdot \frac{y + 6}{y + 6}$

Step 7.3

Multiply $\frac{1}{2}$ by $\frac{y + 6}{y + 6}$ .

$\frac{y + 6}{2 (y + 6)}$

Step 7.4

Cancel the common factor of $y + 6$ .

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Step 7.4.1

Cancel the common factor.

$\frac{y + 6}{2 (y + 6)}$

Step 7.4.2

Rewrite the expression.

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$