Question

The roots of the equation $\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15}$ are
(a) $\frac{5}{2},\frac{3}{2}$
(b) $\frac{5}{2},\frac{-3}{2}$
(c) $\frac{-5}{2},\frac{3}{2}$
(d) $\frac{-5}{2},\frac{-3}{2}$

Answer 1

(c) $\frac{-5}{2},\frac{3}{2}$

$$\begin{gathered} \mathrm{Given}: \\ \frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15} \\ \text{⇒}\frac{x^2+{(x+1)}^2}{(x+1)x}=\frac{34}{15} \\ \text{⇒}{x}^2+x^2+2x+1=\frac{34(x^2+x)}{15} \\ \text{⇒ 15}(\text{2}{x}^2+2x+1)=34x^2+34x \\ \Rightarrow 30x^2+30x+15=34x^2+34x \\ \Rightarrow \text{4}{x}^2+4x-15=0 \\ \text{⇒ 4}{x}^2+10x-6x-15=0 \\ \Rightarrow 2x(2x+5)-3(2x+5)=0 \\ \Rightarrow (2x+5)(2x-3)=0 \\ \text{⇒}x=\frac{-5}{2}\text{or}x=\frac{3}{2} \\ \text{Thus, the roots are}\frac{-5}{2}\text{and}\frac{3}{2}\text{.} \end{gathered}$$