Question

Simplify:$\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{w^2+9w+14}$.

Short answer

The simplified expression is$\frac{3w}{w+7}$.

Step-by-step solution

Step 1. Given Information

The given expression is$\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{w^2+9w+14}$.

Step 2. Express with common denominator   

• Factor the denominators of the given expression.

$\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{(w+2)(w+7)}$

• The LCD of the terms of obtained expression is $(w+2)(w+7)$.
• Multiply the numerator and denominator of first term by $(w+7)$and second term by $(w+2)$.

role="math" localid="1645095547915" $\frac{3w(w+7)}{(w+2)(w+7)}+\frac{2(w+2)}{(w+7)(w+2)}-\frac{17w+4}{(w+2)(w+7)}=\frac{3w^2+21w}{(w+2)(w+7)}+\frac{2w+4}{(w+7)(w+2)}-\frac{17w+4}{(w+2)(w+7)}$

Step 3. Simplify

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

$\frac{3w^2+21w+2w+4-17w-4}{(w+2)(w+7)}=\frac{3w^2+6w}{(w+2)(w+7)}$

• Factor the numerator and cancel the common factors from numerator and denominator.

$\frac{3w(w+2)}{(w+2)(w+7)}=\frac{3w}{w+7}$

• So, the resultant expression is $\frac{3w}{w+7}$.