Example 17. Solve the equation

$$
\frac{2}{2-\sqrt{x}}+\frac{1}{2}=\frac{4}{2 \sqrt{x}-x}
$$
Solution. The domain of the equation is determined by the system of inequalities

$$
\left\{\begin{array}{r}
x \geqslant 0 \\
2-\sqrt{x} \neq 0 \\
2 \sqrt{-} x-x \neq 0
\end{array}\right.
$$

i.e., $x>0, x \neq 4$.

Multiplying both sides of the original equation by the expression $2(2 \sqrt{ } \bar{x}-x)$, we obtain the equation

$$
4 \sqrt{ } \bar{x}+2 \sqrt{x}-x=8
$$

which is a consequence of the original equation (equation (5) has a wider domain: $x \geqslant 0$).

Solving equation (5), we find two roots: $x_{1}=16$ and $x_{2}=4$. The number $x_{2}=4$ does not belong to the domain of equation (4) and is therefore an extraneous root. Substituting the number $x_{1}=16$ into equation (4) shows that it is the only root of the equation.