<s>[INST] Find all $x \in \mathbb R$ such that $x + \sqrt{3}$ and $x^2 + \sqrt{3}$ are rational. Find all $x \in \mathbb R$ such that $x + \sqrt{3}$ and $x^2 + \sqrt{3}$ are rational. I have started by assuming that $x + \sqrt{3} = \frac{a}{b}$ and substituting $x = \frac{a}{b} - \sqrt{3}$ into $x^2 + \sqrt{3}$. It led me to $\frac{a^2}{b^2} + 3 + \sqrt{3} \cdot \frac{b - 2a}{b}$. I don't know how to continue.
 [/INST] Well, since $x^2 + \sqrt{3}$ is assumed to be rational, then your work implies that
$$b - 2a = 0.$$
Make sure that you understand why this is the case. Once you have this, $2a = b$ and
$$x = \frac 1 2 - \sqrt{3}$$
is the only solution.
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