Question

In the following exercises, solve

$\frac{p}{p+7}=\frac{98}{p^2-49}+8$

Short answer

The solution is$p=6,-7.$

Step-by-step solution

Step 1. Given Information

The given rational expression is$\frac{p}{p+7}=\frac{98}{p^2-49}+8$

Step 2. Least Common Denominator

The LCD of the equation is $(p+7)(p-7)$

Multiply the equation by the LCD,

$$\begin{gathered} (p+7)(p-7)\frac{p}{p+7}=\left[\frac{98}{p^2-49}+8\right](p+7)(p-7) \\ p(p-7)=98+8(p+7)(p-7) \\ {p}^2-7p=98+8(p^2-49) \\ {p}^2-7p=98+8p^2-392 \\ 7p^2+7p-294=0 \\ 7(p^2+p-42)=0 \end{gathered}$$

Step 3. Solution of the equation

On simplifying,

$$\begin{gathered} 7\left(p^2+p-42\right)=0 \\ 7\left(p^2+7p-6p-42\right)=0 \\ 7(p+7)(p-6)=0 \\ Therefore, \\ p+7=0;p-6=0 \\ p=-7,p=6 \end{gathered}$$