Basic Math Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{y^{2} - 12 y + 6^{2}}{y^{2} - 4 y - 12} \cdot \frac{2}{y^{2} - 36}$

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.

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

Step 2.3

Rewrite the polynomial.

$\frac{y^{2} - 2 \cdot y \cdot 6 + 6^{2}}{y^{2} - 4 y - 12} \cdot \frac{2}{y^{2} - 36}$

Step 2.4

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

$\frac{{(y - 6)}^{2}}{y^{2} - 4 y - 12} \cdot \frac{2}{y^{2} - 36}$

Step 3

Factor $y^{2} - 4 y - 12$ 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 $- 4$ .

$- 6, 2$

Step 3.2

Write the factored form using these integers.

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

Step 4

Cancel the common factor of ${(y - 6)}^{2}$ and $y - 6$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$\frac{y - 6}{y + 2} \cdot \frac{2}{y^{2} - 36}$

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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 $b = 6$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $y - 6$ .

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Step 6.1.1

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

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

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

Step 6.2

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

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