Example 13. Solve the equation

$$
2 \sqrt{1-x^{2}}=x-2
$$
Solution. Solving the inequality $1-x^{2} \geqslant 0$, we find the domain of the equation: $-1 \leqslant x \leqslant 1$.

By squaring both sides of the equation and combining like terms, we obtain the equation

$$
5 x^{2}-4 x=0
$$

which is a consequence of the original equation. The roots of this equation are: $x_{1}=0$ and $x_{2}=4 / 5$.

Both of these roots belong to the domain of the original equation. However, for each of them, the right-hand side of the squared equation $2 \sqrt{1-x^{2}}=x-2$ is a negative quantity, while the left-hand side is positive. Therefore, both of these roots are extraneous for the original equation.

Thus, the original equation has no solutions.