Task 5. (20 points) Solve the equation $674 \frac{1}{3}\left(x^{2}+\frac{x^{2}}{(1-x)^{2}}\right)=2023$.
Solution.

$$
\begin{aligned}
& x^{2}+\frac{x^{2}}{(1-x)^{2}}=2023: 674 \frac{1}{3} \\
& x^{2}+\frac{x^{2}}{(1-x)^{2}}=3 \\
& x^{2}-\frac{2 x^{2}}{1-x}+\frac{x^{2}}{(1-x)^{2}}=3-\frac{2 x^{2}}{1-x}
\end{aligned}
$$

$\left(x-\frac{x}{1-x}\right)^{2}=3-\frac{2 x^{2}}{1-x}$

$\left(\frac{x^{2}}{1-x}\right)^{2}=3-2 \cdot \frac{x^{2}}{1-x}$

Let $\frac{x^{2}}{1-x}=y$.

We get the equation $y^{2}=3-2 y$.

$y^{2}+2 y-3=0 ; y=-3$ or $y=1$

1) if $y=-3$, then $\frac{x^{2}}{1-x}=-3 ; x^{2}-3 x+3=0-$ the equation has no real roots.
2) if $y=1$, then $\frac{x^{2}}{1-x}=1 ; x^{2}+x-1=0 ; x=-\frac{1+\sqrt{5}}{2}$ or $x=\frac{\sqrt{5}-1}{2}$.

Answer. $-\frac{1+\sqrt{5}}{2} ; \frac{\sqrt{5}-1}{2}$.