Question

$\frac{1}{(x-3)}-\frac{1}{(x+5)}=\frac{1}{6},(x\ne 3,-5)$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \frac{1}{(x-3)}-\frac{1}{(x+5)}=\frac{1}{6} \\ \Rightarrow \frac{(x+5)-(x-3)}{(x-3)(x+5)}=\frac{1}{6} \\ \Rightarrow \frac{8}{(x^2-2x-15)}=\frac{1}{6} \\ \Rightarrow (x^2-2x-15)=48[\mathrm{On}\text{cross multiplying}] \\ \Rightarrow x^2-2x-63=0 \\ \Rightarrow x^2-(9x-7x)-63=0 \\ \Rightarrow x^2-9x+7x-63=0 \\ \Rightarrow x(x-9)+7(x-9)=0 \\ \Rightarrow (x+7)(x-9)=0 \\ \Rightarrow x+7=0\text{or}x-9=0 \\ \Rightarrow x=-7\text{or}x=9 \\ \mathrm{H}\text{ence, the roots of the equation are}-\text{7 and 9}. \end{gathered}$$