Originally Posted by
mr fantastic The correct equation is $\displaystyle \sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{13}{6}$.
Square both sides:
$\displaystyle \frac{x}{x-1} + 2 + \frac{x-1}{x} =\frac{169}{36}$.
Multiply both sides by $\displaystyle 36 (x - 1) x$:
$\displaystyle 36 x^2 + 72 x (x - 1) + 36 (x - 1)^2 = 169 x (x - 1)$
Expand and simplify into a routine quadratic equation. Solve the quadratic equation. Substitute the solutions into the original equation and reject the ones (if any) that fail to satisfy it.