Question

In the following exercises, perform indicated operations.

$\frac{-2r-16}{r^2+6r-16}-\frac{5}{2-r}$

Short answer

The resultant expression is$\frac{3}{r-2}$.

Step-by-step solution

Step 1. Given Information

The given expression is$\frac{-2r-16}{r^2+6r-16}-\frac{5}{2-r}$.

Step 2. Express with LCD

• The LCD of the given expressions to subtract is $\left(r+8\right)\left(r-2\right)$.
• Express the given expression with LCD.

$\frac{-2r-16}{r^2+6r-16}-\frac{5}{2-r}=\frac{-2r-16}{\left(r+8\right)\left(r-2\right)}-\frac{-5r-40}{\left(r+8\right)\left(r-2\right)}$

Step 3. Simplify

• As the denominator is the same for both the terms, perform subtraction in the numerator and place it over the common denominator.

$\frac{-2r-16+5r+40}{\left(r+8\right)\left(r-2\right)}=\frac{3r+24}{\left(r+8\right)\left(r-2\right)}$

• Factor the numerator and cancel out the common factor from numerator and denominator.

$\frac{3\left(r+8\right)}{\left(r+8\right)\left(r-2\right)}=\frac{3}{r-2}$

• So, the resultant expression is$\frac{3}{r-2}$