<s>[INST] Solving exponential equation of form $(a+b)^x + (a-b)^x = c$ I'm given equation:
$$\left(2-\sqrt3\right)^\frac{x}{2} + \left(2+\sqrt3\right)^\frac{x}{2} = 4,$$
and I'm stuck with it. Which way shall I dig into to solve it?
 [/INST] Let $u = (2+\sqrt{3})^{x/2}$. Using the observation that $2-\sqrt{3} = \frac{1}{2+\sqrt{3}}$, we have that $(2-\sqrt{3})^{x/2} + (2+\sqrt{3})^{x/2} = u+\frac{1}{u}$. We thus aim to solve
$$ u + \frac{1}{u} = 4. $$
Since $u\ne 0$, this is equivalent to $u^2 - 4u + 1 = 0$. Using the quadratic formula, we obtain $u = 2\pm\sqrt{3}$. We thus have
$$ (2+\sqrt{3})^{\frac{x}{2}} = 2\pm\sqrt{3} = (2+\sqrt{3})^{\pm 1}\implies \boxed{x = \pm 2}. $$
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