Example 32. Solve the equation

$$
\sqrt{1-x^{2}}=\left(\frac{2}{3}-\sqrt{\bar{x}}\right)^{4}
$$
Solution. Let $\sqrt{x}=u \quad$ and $\quad 2 / 3-\sqrt{x}=v, \quad$ then we have $1-x^{2}=1-u^{4}$ and $(2 / 3-\sqrt{\bar{x}})^{4}=v^{4}$. For $u$ and $v$, we get:

$$
\left\{\begin{array} { r l } 
{ u ^ { 4 } + v ^ { 4 } } & { = 1 } \\
{ u + v } & { = 2 / 3 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
2(u v)^{2}-\frac{16}{9} u v-\frac{65}{81}=0 \\
u+v=2 / 3
\end{array}\right.\right.
$$

This system is equivalent to the combination of two systems:

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-073.jpg?height=194&width=862&top_left_y=882&top_left_x=583)

The first system in the combination (22) has no solutions, as the discriminant of the quadratic equation $a^{2}-\frac{2}{3} a+\frac{8+\sqrt{194}}{18}=0$ is negative.

The second system in the combination (22) has two solutions:

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-073.jpg?height=280&width=1208&top_left_y=1322&top_left_x=404)

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-073.jpg?height=68&width=1351&top_left_y=1602&top_left_x=358)
which is equivalent to the equation

$$
\sqrt{\bar{x}}=\frac{1}{3}\left(1+\sqrt{\sqrt{\frac{97}{2}}-3}\right)
$$

from which we get $x_{1}=\frac{1}{9}\left(1+\sqrt{\sqrt{\frac{97}{2}}-3}\right)^{2}-$ the only root of equation (21).