Question

In the following exercises, perform indicated operations.

$\frac{2t-30}{t^2+6t-27}-\frac{2}{3-t}$

Short answer

The resultant expression is$\frac{4}{t+9}$.

Step-by-step solution

Step 1. Given Information

The given expression is$\frac{2t-30}{t^2+6t-27}-\frac{2}{3-t}$.

Step 2. Express with LCD

• The LCD of the given terms is $\left(t+9\right)\left(t-3\right)$.
• Express the given expression with the LCD.

localid="1646140499486" $\frac{2t-30}{t^2+6t-27}-\frac{2}{3-t}=\frac{2t-30}{\left(t+9\right)\left(t-3\right)}-\frac{-2t+18}{\left(t+9\right)\left(t-3\right)}$

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.

$\frac{2t-30+2t-18}{\left(t+9\right)\left(t-3\right)}=\frac{4t-12}{\left(t+9\right)\left(t-3\right)}$

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

$\frac{4\left(t-3\right)}{\left(t+9\right)\left(t-3\right)}=\frac{4}{t+9}$

• So, the resultant expression is$\frac{4}{t+9}$