Precalculus Examples

$\frac{2}{r + 10} - \frac{4}{r - 6} - \frac{r}{r^{2} + 4 r - 60}$

Step 1

Factor $r^{2} + 4 r - 60$ using the AC method.

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Step 1.1

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 $- 60$ and whose sum is $4$ .

$- 6, 10$

Step 1.2

Write the factored form using these integers.

$\frac{2}{r + 10} - \frac{4}{r - 6} - \frac{r}{(r - 6) (r + 10)}$

Step 2

To write $\frac{2}{r + 10}$ as a fraction with a common denominator, multiply by $\frac{r - 6}{r - 6}$ .

$\frac{2}{r + 10} \cdot \frac{r - 6}{r - 6} - \frac{4}{r - 6} - \frac{r}{(r - 6) (r + 10)}$

Step 3

To write $- \frac{4}{r - 6}$ as a fraction with a common denominator, multiply by $\frac{r + 10}{r + 10}$ .

$\frac{2}{r + 10} \cdot \frac{r - 6}{r - 6} - \frac{4}{r - 6} \cdot \frac{r + 10}{r + 10} - \frac{r}{(r - 6) (r + 10)}$

Step 4

Write each expression with a common denominator of $(r + 10) (r - 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{r + 10}$ by $\frac{r - 6}{r - 6}$ .

$\frac{2 (r - 6)}{(r + 10) (r - 6)} - \frac{4}{r - 6} \cdot \frac{r + 10}{r + 10} - \frac{r}{(r - 6) (r + 10)}$

Step 4.2

Multiply $\frac{4}{r - 6}$ by $\frac{r + 10}{r + 10}$ .

$\frac{2 (r - 6)}{(r + 10) (r - 6)} - \frac{4 (r + 10)}{(r - 6) (r + 10)} - \frac{r}{(r - 6) (r + 10)}$

Step 4.3

Reorder the factors of $(r - 6) (r + 10)$ .

$\frac{2 (r - 6)}{(r + 10) (r - 6)} - \frac{4 (r + 10)}{(r + 10) (r - 6)} - \frac{r}{(r - 6) (r + 10)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{2 (r - 6) - 4 (r + 10)}{(r + 10) (r - 6)} - \frac{r}{(r - 6) (r + 10)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{2 (r - 6) - 4 (r + 10) - r}{(r + 10) (r - 6)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{2 r + 2 \cdot - 6 - 4 (r + 10) - r}{(r + 10) (r - 6)}$

Step 6.2

Multiply $2$ by $- 6$ .

$\frac{2 r - 12 - 4 (r + 10) - r}{(r + 10) (r - 6)}$

Step 6.3

Apply the distributive property.

$\frac{2 r - 12 - 4 r - 4 \cdot 10 - r}{(r + 10) (r - 6)}$

Step 6.4

Multiply $- 4$ by $10$ .

$\frac{2 r - 12 - 4 r - 40 - r}{(r + 10) (r - 6)}$

Step 7

Simplify terms.

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Step 7.1

Subtract $4 r$ from $2 r$ .

$\frac{- 2 r - 12 - 40 - r}{(r + 10) (r - 6)}$

Step 7.2

Subtract $r$ from $- 2 r$ .

$\frac{- 3 r - 12 - 40}{(r + 10) (r - 6)}$

Step 7.3

Subtract $40$ from $- 12$ .

$\frac{- 3 r - 52}{(r + 10) (r - 6)}$

Step 7.4

Factor $- 1$ out of $- 3 r$ .

$\frac{- (3 r) - 52}{(r + 10) (r - 6)}$

Step 7.5

Rewrite $- 52$ as $- 1 (52)$ .

$\frac{- (3 r) - 1 (52)}{(r + 10) (r - 6)}$

Step 7.6

Factor $- 1$ out of $- (3 r) - 1 (52)$ .

$\frac{- (3 r + 52)}{(r + 10) (r - 6)}$

Step 7.7

Simplify the expression.

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Step 7.7.1

Rewrite $- (3 r + 52)$ as $- 1 (3 r + 52)$ .

$\frac{- 1 (3 r + 52)}{(r + 10) (r - 6)}$

Step 7.7.2

Move the negative in front of the fraction.

$- \frac{3 r + 52}{(r + 10) (r - 6)}$