Question

In the following exercises, add or subtract.

$\frac{a^2+3a}{a^2-9}-\frac{3a-27}{9-a^2}$

Short answer

The required expression is$\frac{a+9}{a+3}$.

Step-by-step solution

Step 1. Given Information

The given expression is$\frac{a^2+3a}{a^2-9}-\frac{3a-27}{9-a^2}$.

Step 2. Multiply the numerator and denominator of the second expression by -1

• Multiply the numerator and denominator of the second term by$-1$ to make the denominators equal.

$\frac{a^2+3a}{a^2-9}-\frac{3a-27}{9-a^2}=\frac{a^2+3a}{a^2-9}-\frac{-3a+27}{a^2-9}$

Step 3. Subtract the numerators

• As the denominators of the rational expressions is equal, subtract the numerators and place them over a common denominator.

$\begin{array}{rcl}\frac{a^2+3a}{a^2-9}-\frac{-3a-27}{a^2-9}& =& \frac{a^2+3a-\left(-3a+27\right)}{a^2-9}\\ & =& \frac{a^2+3a+3a-27}{a^2-9}\\ & =& \frac{{\displaystyle a^2+6a-27}}{a^2-9}\end{array}$

Step 4. Factor the numerator and denominator 

• Factor the expression in the numerator and denominator and then cancel out the common terms in numerator and denominator.