Why is $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ rational? For this question,

Show the following irrational-looking expressions are actually rational numbers.
(a) $\sqrt{4+2\sqrt{3}}-\sqrt{3}$, and
(b) ...

I solved it as follows:
$$\begin{align} x &= \sqrt{4+2\sqrt{3}}-\sqrt{3},\\
x+\sqrt{3} &= \sqrt{4+2\sqrt{3}},\\
(x+\sqrt{3})^{2} &= (\sqrt{4+2\sqrt{3}})^{2},\\
\end{align}\\
x^{2}+2\sqrt{3}x-(1+2\sqrt{3}) = 0,\\
(x-1)(x+(1+2\sqrt{3}))=0.$$
My question is that, there are two numbers satisfying $x = \sqrt{4+2\sqrt{3}}-\sqrt{3}$, but one of them is irrational. Then, how can we say $\sqrt{4+2\sqrt{3}}-\sqrt{3}$ is rational as a whole?

$\sqrt{4+2\sqrt{3}}=\sqrt{1+2\sqrt{3}+3}=\sqrt{(1+\sqrt{3})^2}=1+\sqrt{3}$
