Algebra Examples

$- \frac{2}{n + 4} - \frac{n^{2}}{n^{2} - 16}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $16$ as $4^{2}$ .

$- \frac{2}{n + 4} - \frac{n^{2}}{n^{2} - 4^{2}}$

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 $a = n$ and $b = 4$ .

$- \frac{2}{n + 4} - \frac{n^{2}}{(n + 4) (n - 4)}$

Step 2

To write $- \frac{2}{n + 4}$ as a fraction with a common denominator, multiply by $\frac{n - 4}{n - 4}$ .

$- \frac{2}{n + 4} \cdot \frac{n - 4}{n - 4} - \frac{n^{2}}{(n + 4) (n - 4)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2}{n + 4}$ by $\frac{n - 4}{n - 4}$ .

$- \frac{2 (n - 4)}{(n + 4) (n - 4)} - \frac{n^{2}}{(n + 4) (n - 4)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{- 2 (n - 4) - n^{2}}{(n + 4) (n - 4)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{- 2 n - 2 \cdot - 4 - n^{2}}{(n + 4) (n - 4)}$

Step 4.2

Multiply $- 2$ by $- 4$ .

$\frac{- 2 n + 8 - n^{2}}{(n + 4) (n - 4)}$

Step 4.3

Reorder terms.

$\frac{- n^{2} - 2 n + 8}{(n + 4) (n - 4)}$

Step 4.4

Factor by grouping.

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Step 4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

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Step 4.4.1.1

Factor $- 2$ out of $- 2 n$ .

$\frac{- n^{2} - 2 (n) + 8}{(n + 4) (n - 4)}$

Step 4.4.1.2

Rewrite $- 2$ as $2$ plus $- 4$

$\frac{- n^{2} + (2 - 4) n + 8}{(n + 4) (n - 4)}$

Step 4.4.1.3

Apply the distributive property.

$\frac{- n^{2} + 2 n - 4 n + 8}{(n + 4) (n - 4)}$

Step 4.4.2

Factor out the greatest common factor from each group.

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Step 4.4.2.1

Group the first two terms and the last two terms.

$\frac{(- n^{2} + 2 n) - 4 n + 8}{(n + 4) (n - 4)}$

Step 4.4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{n (- n + 2) + 4 (- n + 2)}{(n + 4) (n - 4)}$

Step 4.4.3

Factor the polynomial by factoring out the greatest common factor, $- n + 2$ .

$\frac{(- n + 2) (n + 4)}{(n + 4) (n - 4)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $n + 4$ .

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Step 5.1.1

Cancel the common factor.

$\frac{(- n + 2) (n + 4)}{(n + 4) (n - 4)}$

Step 5.1.2

Rewrite the expression.

$\frac{- n + 2}{n - 4}$

Step 5.2

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

$\frac{- (n) + 2}{n - 4}$

Step 5.3

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- (n) - 1 (- 2)}{n - 4}$

Step 5.4

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

$\frac{- (n - 2)}{n - 4}$

Step 5.5

Simplify the expression.

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Step 5.5.1

Rewrite $- (n - 2)$ as $- 1 (n - 2)$ .

$\frac{- 1 (n - 2)}{n - 4}$

Step 5.5.2

Move the negative in front of the fraction.

$- \frac{n - 2}{n - 4}$