Basic Math Examples

$\frac{- 2}{6 - n} + \frac{13}{n^{2} - 36}$

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

$- \frac{2}{6 - n} + \frac{13}{n^{2} - 36}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$- \frac{2}{6 - n} + \frac{13}{n^{2} - 6^{2}}$

Step 1.2.2

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

$- \frac{2}{6 - n} + \frac{13}{(n + 6) (n - 6)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $n$ .

$- \frac{2}{6 - n} + \frac{13}{(n + 6) (- 1 (- n) - 6)}$

Step 2.2

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

$- \frac{2}{6 - n} + \frac{13}{(n + 6) (- 1 (- n) - 1 (6))}$

Step 2.3

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

$- \frac{2}{6 - n} + \frac{13}{(n + 6) (- 1 (- n + 6))}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{13}{(n + 6) (- 1 (- n + 6))}$ to the numerator.

$- \frac{2}{6 - n} + \frac{- 1 \cdot 13}{(n + 6) (- n + 6)}$

Step 2.4.2

Reorder terms.

$- \frac{2}{- n + 6} + \frac{- 1 \cdot 13}{(n + 6) (- n + 6)}$

Step 3

To write $- \frac{2}{- n + 6}$ as a fraction with a common denominator, multiply by $\frac{n + 6}{n + 6}$ .

$- \frac{2}{- n + 6} \cdot \frac{n + 6}{n + 6} + \frac{- 1 \cdot 13}{(n + 6) (- n + 6)}$

Step 4

Write each expression with a common denominator of $(- n + 6) (n + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{- n + 6}$ by $\frac{n + 6}{n + 6}$ .

$- \frac{2 (n + 6)}{(- n + 6) (n + 6)} + \frac{- 1 \cdot 13}{(n + 6) (- n + 6)}$

Step 4.2

Reorder the factors of $(n + 6) (- n + 6)$ .

$- \frac{2 (n + 6)}{(- n + 6) (n + 6)} + \frac{- 1 \cdot 13}{(- n + 6) (n + 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 2 (n + 6) - 1 \cdot 13}{(- n + 6) (n + 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $- 1$ out of $- 2 (n + 6) - 1 \cdot 13$ .

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Step 6.1.1

Factor $- 1$ out of $- 2 (n + 6)$ .

$\frac{- (2 (n + 6)) - 1 \cdot 13}{(- n + 6) (n + 6)}$

Step 6.1.2

Factor $- 1$ out of $- 1 \cdot 13$ .

$\frac{- (2 (n + 6)) - 1 (13)}{(- n + 6) (n + 6)}$

Step 6.1.3

Factor $- 1$ out of $- (2 (n + 6)) - 1 (13)$ .

$\frac{- (2 (n + 6) + 13)}{(- n + 6) (n + 6)}$

Step 6.2

Apply the distributive property.

$\frac{- (2 n + 2 \cdot 6 + 13)}{(- n + 6) (n + 6)}$

Step 6.3

Multiply $2$ by $6$ .

$\frac{- (2 n + 12 + 13)}{(- n + 6) (n + 6)}$

Step 6.4

Add $12$ and $13$ .

$\frac{- (2 n + 25)}{(- n + 6) (n + 6)}$

Step 7

Simplify with factoring out.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{2 n + 25}{(- n + 6) (n + 6)}$

Step 7.2

Factor $- 1$ out of $- n$ .

$- \frac{2 n + 25}{(- (n) + 6) (n + 6)}$

Step 7.3

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

$- \frac{2 n + 25}{(- (n) - 1 (- 6)) (n + 6)}$

Step 7.4

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

$- \frac{2 n + 25}{- (n - 6) (n + 6)}$

Step 7.5

Simplify the expression.

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Step 7.5.1

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

$- \frac{2 n + 25}{- 1 (n - 6) (n + 6)}$

Step 7.5.2

Move the negative in front of the fraction.

$- - \frac{2 n + 25}{(n - 6) (n + 6)}$

Step 7.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{2 n + 25}{(n - 6) (n + 6)}$

Step 7.5.4

Multiply $\frac{2 n + 25}{(n - 6) (n + 6)}$ by $1$ .

$\frac{2 n + 25}{(n - 6) (n + 6)}$