Basic Math Examples

$\frac{36 - y^{2}}{y^{2} + 5 y - 6} \cdot \frac{y^{2} + 7 + 10}{y^{2} - 4 y - 12}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{6^{2} - y^{2}}{y^{2} + 5 y - 6} \cdot \frac{y^{2} + 7 + 10}{y^{2} - 4 y - 12}$

Step 1.2

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

$\frac{(6 + y) (6 - y)}{y^{2} + 5 y - 6} \cdot \frac{y^{2} + 7 + 10}{y^{2} - 4 y - 12}$

Step 2

Factor $y^{2} + 5 y - 6$ 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 $5$ .

$- 1, 6$

Step 2.2

Write the factored form using these integers.

$\frac{(6 + y) (6 - y)}{(y - 1) (y + 6)} \cdot \frac{y^{2} + 7 + 10}{y^{2} - 4 y - 12}$

Step 3

Add $7$ and $10$ .

$\frac{(6 + y) (6 - y)}{(y - 1) (y + 6)} \cdot \frac{y^{2} + 17}{y^{2} - 4 y - 12}$

Step 4

Factor $y^{2} - 4 y - 12$ using the AC method.

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Step 4.1

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 4.2

Write the factored form using these integers.

$\frac{(6 + y) (6 - y)}{(y - 1) (y + 6)} \cdot \frac{y^{2} + 17}{(y - 6) (y + 2)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $6 + y$ and $y + 6$ .

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Step 5.1.1

Reorder terms.

$\frac{(y + 6) (6 - y)}{(y - 1) (y + 6)} \cdot \frac{y^{2} + 17}{(y - 6) (y + 2)}$

Step 5.1.2

Cancel the common factor.

$\frac{(y + 6) (6 - y)}{(y - 1) (y + 6)} \cdot \frac{y^{2} + 17}{(y - 6) (y + 2)}$

Step 5.1.3

Rewrite the expression.

$\frac{6 - y}{y - 1} \cdot \frac{y^{2} + 17}{(y - 6) (y + 2)}$

Step 5.2

Multiply $\frac{6 - y}{y - 1}$ by $\frac{y^{2} + 17}{(y - 6) (y + 2)}$ .

$\frac{(6 - y) (y^{2} + 17)}{(y - 1) ((y - 6) (y + 2))}$

Step 5.3

Cancel the common factor of $6 - y$ and $y - 6$ .

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Step 5.3.1

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{(- 1 (- 6) - y) (y^{2} + 17)}{(y - 1) ((y - 6) (y + 2))}$

Step 5.3.2

Factor $- 1$ out of $- y$ .

$\frac{(- 1 (- 6) - (y)) (y^{2} + 17)}{(y - 1) ((y - 6) (y + 2))}$

Step 5.3.3

Factor $- 1$ out of $- 1 (- 6) - (y)$ .

$\frac{- 1 (- 6 + y) (y^{2} + 17)}{(y - 1) ((y - 6) (y + 2))}$

Step 5.3.4

Reorder terms.

$\frac{- 1 (y - 6) (y^{2} + 17)}{(y - 1) (y - 6) (y + 2)}$

Step 5.3.5

Cancel the common factor.

$\frac{- 1 (y - 6) (y^{2} + 17)}{(y - 1) (y - 6) (y + 2)}$

Step 5.3.6

Rewrite the expression.

$\frac{- 1 (y^{2} + 17)}{(y - 1) (y + 2)}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{y^{2} + 17}{(y - 1) (y + 2)}$