Basic Math Examples

$\frac{3}{y + 6} + \frac{36}{y^{2} - 36} - \frac{8}{y - 6}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{3}{y + 6} + \frac{36}{y^{2} - 6^{2}} - \frac{8}{y - 6}$

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 $a = y$ and $b = 6$ .

$\frac{3}{y + 6} + \frac{36}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 2

To write $\frac{3}{y + 6}$ as a fraction with a common denominator, multiply by $\frac{y - 6}{y - 6}$ .

$\frac{3}{y + 6} \cdot \frac{y - 6}{y - 6} + \frac{36}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3}{y + 6}$ by $\frac{y - 6}{y - 6}$ .

$\frac{3 (y - 6)}{(y + 6) (y - 6)} + \frac{36}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 (y - 6) + 36}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 4

Simplify each term.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $3$ out of $3 (y - 6) + 36$ .

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Step 4.1.1.1

Factor $3$ out of $36$ .

$\frac{3 (y - 6) + 3 \cdot 12}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 4.1.1.2

Factor $3$ out of $3 (y - 6) + 3 \cdot 12$ .

$\frac{3 (y - 6 + 12)}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 4.1.2

Add $- 6$ and $12$ .

$\frac{3 (y + 6)}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 4.2

Cancel the common factor of $y + 6$ .

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Step 4.2.1

Cancel the common factor.

$\frac{3 (y + 6)}{(y + 6) (y - 6)} - \frac{8}{y - 6}$

Step 4.2.2

Rewrite the expression.

$\frac{3}{y - 6} - \frac{8}{y - 6}$

Step 5

Combine fractions.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{3 - 8}{y - 6}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Subtract $8$ from $3$ .

$\frac{- 5}{y - 6}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{5}{y - 6}$