Question

Solve each of the following quadratic equations:

$\frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7},x\ne 1,-5$ [CBSE 2010]

Answer 1

$$\begin{gathered} \frac{1}{x-1}-\frac{1}{x+5}=\frac{6}{7},x\ne 1,-5 \\ \Rightarrow \frac{x+5-x+1}{\left(x-1\right)\left(x+5\right)}=\frac{6}{7} \\ \Rightarrow \frac{6}{x^2+4x-5}=\frac{6}{7} \\ \Rightarrow x^2+4x-5=7 \end{gathered}$$
$$\begin{gathered} \Rightarrow x^2+4x-12=0 \\ \Rightarrow x^2+6x-2x-12=0 \\ \Rightarrow x\left(x+6\right)-2\left(x+6\right)=0 \\ \Rightarrow \left(x+6\right)\left(x-2\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x+6=0\mathrm{or}x-2=0 \\ \Rightarrow x=-6\mathrm{or}x=2 \end{gathered}$$
Hence, −6 and 2 are the roots of the given equation.