Example 17. Solve the equation:
$$
\left(x-\frac{1}{x}\right)^{\frac{1}{2}}+\left(1-\frac{1}{x}\right)^{\frac{1}{2}}=x .
$$
Multiply both sides of (1) by the conjugate radical $\left(x-\frac{1}{x}\right)^{\frac{1}{2}}-\left(1-\frac{1}{x}\right)^{\frac{1}{2}}$, and simplify to get
$$
\left(x-\frac{1}{x}\right)^{\frac{1}{2}}-\left(1-\frac{1}{x}\right)^{\frac{1}{2}}=1-\frac{1}{x} \text {. }
$$

By (1) + (2), we get
$2\left(x-\frac{1}{x}\right)^{\frac{1}{2}}=1+x-\frac{1}{x}$,
which means
$$
\begin{array}{l}
\left(x-\frac{1}{x}\right)-2\left(x-\frac{1}{x}\right)^{\frac{1}{2}}+1=0, \\
\left(x-\frac{1}{x}\right)^{\frac{1}{2}}=1 .
\end{array}
$$

Simplifying and rearranging, we get $x^{2}-x-1=0$,
solving it, we get $\mathrm{x}=\frac{1 \pm \sqrt{ } 5}{2}$.
Upon inspection, $x=\frac{1-\sqrt{5}}{2}$ is an extraneous root, and $x=\frac{\sqrt{5}+1}{2}$ is the root of the original equation.