Question

Solve the following equations by reducing them in quadractic form.
$\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{13}{6},x\ne 0,1$

Answer 1

$\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{13}{6}$
Squaring both sides,
$\frac{x}{x-1}+\frac{x-1}{x}+2=\frac{{13}^2}{6^2}\frac{x^2+(x-1{)}^2}{x(x-1)}=\frac{169}{36}-2=\frac{97}{36}=36(x^2+x2-2x+1)=98(x^2-x)=97x^2-x^2-97x+72x-36=025x^2-25x-36=0$
Solving
$x=\frac{25\pm \sqrt{{25}^2+4\times 35\times 5}}{50}=\frac{9}{5}$ or $-\frac{4}{5}$