Problem 2. Determine the irrational numbers $\mathrm{x}$, with the property that the numbers $\mathrm{x}^{2}+\mathrm{x}$ and

$$
x^{3}+2 x^{2} \text { are integers. }
$$

Mathematical Gazette
## Solution.

Let $a=x^{2}+x, a \in Z$. We have $x^{3}+2 x^{2}=x^{2}(x+2)=(a-x)(x+2)=-x^{2}+(a-2) x+2 a=(a-1) x+a(1)$.

Since $x^{3}+2 x^{2}, a, a-1$ are integers, and $x$ is irrational, we get $a-1=0 \Rightarrow a=1$.

Substituting into relation (1) we obtain the equation: $x^{2}+x-1=0$

(1p)

Solving the equation we get $x \in\left\{\frac{-1 \pm \sqrt{5}}{2}\right\}$ and both solutions are valid because they are irrational numbers and satisfy the given property.