Problem 1. Determine the irrational number $x$ knowing that the numbers $x^{2}+x$ and $x^{3}+2 x^{2}$ are integers.
Solution. Let $x^{2}+x=a$ and $x^{3}+2 x^{2}=b$. Then $b-a x=x^{2}=a-x$, so

![](https://cdn.mathpix.com/cropped/2024_06_07_a70accb70efe75559608g-2.jpg?height=48&width=1341&top_left_y=651&top_left_x=392)

Since $x$ is irrational and $a, b$ are integers, we deduce $a=b=1, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots . \ldots 2 p$

![](https://cdn.mathpix.com/cropped/2024_06_07_a70accb70efe75559608g-2.jpg?height=54&width=1298&top_left_y=737&top_left_x=435)