Question

Solve $\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{5}{2}$.

Answer 1

$\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{5}{2}$

$⟹\frac{x+x-1}{\sqrt{x}\sqrt{x-1}}=\frac{5}{2}$ (Multiplying)

$⟹(2x-1)\times 2=5\sqrt{x}\sqrt{x-1}$

squaring

$⟹(2x-1{)}^2\times 2^2=5^{2}x(x-1)$

$⟹(4x^2-4x+1)4=25(x^2-1)$

$⟹16x^2-16x+4=25x^2-25$

$⟹25x^2-16x^2+16x-25-4=0$

$⟹a{x}^2+16x-29=0$

The standard quadratic equation is $a{x}^2+bx+c=0$

$⟹x=\frac{-16\pm \sqrt{{16}^2-4\times 9\times (-29)}}{2\times 9}$ $(x=\frac{-b\pm \sqrt{b^2-4ac}}{2a})$

$x=\frac{-16\pm 10\sqrt{13}}{18}$

$\therefore x=\frac{8+5\sqrt{13}}{9}$ OR $x=\frac{8-5\sqrt{13}}{9}$