Question

Solve:$\frac{x}{x+1}+\frac{x+1}{x}=\frac{34}{15},x\ne -1$ and $x\ne 0$

Answer 1

Let $t=\frac{x+1}{x}$
$t+\frac{1}{t}=\frac{34}{15}$
$⟹t^2+1=\frac{34}{15}t$
$⟹15t^2-34t+15=0$
$⟹15t^2-25t-9t+15=0$
$⟹5t(3t-5)-3(3t-5)=0$
$⟹(5t-3)(3t-5)=0$
$⟹t=\frac{3}{5},\frac{5}{3}$
$⟹\frac{x+1}{x}=\frac{3}{5},\frac{5}{3}$
$⟹1+\frac{1}{x}=\frac{3}{5},\frac{5}{3}$
$⟹\frac{1}{x}=\frac{-2}{5},\frac{2}{3}$
$⟹x=-\frac{5}{2},\frac{3}{2}$