Question

Solve each of the following quadratic equations:

$\frac{x}{x+1}+\frac{x+1}{x}=2\frac{4}{15},x\ne 0,-1$ [CBSE 2014]

Answer 1

$$\begin{gathered} \frac{x}{x+1}+\frac{x+1}{x}=2\frac{4}{15},x\ne 0,-1 \\ \Rightarrow \frac{x^2+{\left(x+1\right)}^2}{x\left(x+1\right)}=\frac{34}{15} \\ \Rightarrow \frac{x^2+x^2+2x+1}{x^2+x}=\frac{34}{15} \\ \Rightarrow \frac{2x^2+2x+1}{x^2+x}=\frac{34}{15} \end{gathered}$$
$$\begin{gathered} \Rightarrow 30x^2+30x+15=34x^2+34x \\ \Rightarrow 4x^2+4x-15=0 \\ \Rightarrow 4x^2+10x-6x-15=0 \\ \Rightarrow 2x\left(2x+5\right)-3\left(2x+5\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow \left(2x+5\right)\left(2x-3\right)=0 \\ \Rightarrow 2x+5=0\mathrm{or}2x-3=0 \\ \Rightarrow x=-\frac{5}{2}\mathrm{or}x=\frac{3}{2} \end{gathered}$$
Hence, $-\frac{5}{2}$ and $\frac{3}{2}$ are the roots of the given equation.