Question

Solve the equation : $\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{13}{6},x\ne 0,1$

Answer 1

We have,

$\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{13}{6}$

On squaring both sides, we get

$\frac{x}{x-1}+\frac{x-1}{x}+2\sqrt{\frac{x}{x-1}\times \frac{x-1}{x}}=\frac{169}{36}$

$\frac{x^2+x^2+1-2x}{x^2-x}+2=\frac{169}{36}$

$\frac{2x^2+1-2x}{x^2-x}=\frac{169}{36}-2$

$\frac{2x^2+1-2x}{x^2-x}=\frac{97}{36}$

$72x^2+36-72x=97x^2-97x$

$25x^2-25x-36=0$

$25x^2-45x+20x-36=0$

$5x(5x-9)+4(5x-9)=0$

$(5x+4)(5x-9)=0$

$x=-\frac{4}{5},\frac{9}{5}$

Hence, this is the answer.