Answer
$\frac{100}{99}$.
Work Step by Step
$\frac{x}{x-1}-6\sqrt{\frac{x}{x-1}}-40=0$.
Let $u=\sqrt{\frac{x}{x-1}}$ and ${{u}^{2}}=\frac{x}{x-1}$. Substitute these values in the expression $\frac{x}{x-1}-6\sqrt{\frac{x}{x-1}}-40=0$.
$\begin{align}
& {{u}^{2}}-6u-40=0 \\
& {{u}^{2}}-10u+4u-40=0 \\
& u\left( u-10 \right)+4\left( u-10 \right)=0 \\
& \left( u-10 \right)\left( u+4 \right)=0
\end{align}$
Thus,
$u=10$ or $u=-4$.
Now, replacing u with $\sqrt{\frac{x}{x-1}}$, we find:
$\sqrt{\frac{x}{x-1}}=10$
$\begin{align}
& {{\left( \sqrt{\frac{x}{x-1}} \right)}^{2}}={{\left( 10 \right)}^{2}} \\
& \frac{x}{x-1}=100 \\
& x=100x-100 \\
& x=\frac{100}{99}
\end{align}$
Therefore, the value of x is $\frac{100}{99}$.
Now, replacing u with $\sqrt{\frac{x}{x-1}}$, we find:
$\sqrt{\frac{x}{x-1}}=-4$, has no solution.
Thus, the solution of the expression $\frac{x}{x-1}-6\sqrt{\frac{x}{x-1}}-40=0$ is $\frac{100}{99}$.
