<s>[INST] Define $a$ and $b$ rational numbers so they satisfy equation Define $a, b \in \mathbb{Q}$ so that
$$\frac{a}{\sqrt{7 + 4\sqrt{3}}} + \frac{b}{\sqrt{7 - 4\sqrt{3}}} = \sqrt{4 + 2\sqrt{3}}$$
Using $\sqrt{a \pm \sqrt{b}} = \sqrt{\frac{a + \sqrt{a^2 - b}}{2}} \pm \sqrt{\frac{a - \sqrt{a^2 - b}}{2}}$ I got $\frac{a}{2 + \sqrt{3}} + \frac{b}{2 - \sqrt{3}} = \sqrt{3}+1$ which results in $2(a+b)+\sqrt{3}(b-a) = \sqrt{3}+1$, I'm not sure how to proceed from that.
 [/INST] Hint:
$$\frac{1}{\sqrt{7+4\sqrt{3}}}=2-\sqrt{3}$$
$$\frac{1}{\sqrt{7-4\sqrt{3}}}=2+\sqrt{3}$$
$$\sqrt{4+2\sqrt{3}}=1+\sqrt{3}$$
</s>