The main question is : If $(\sqrt{x^2-5x+6} + \sqrt{x^2-5x+4})^{x/2} + (\sqrt{x^2-5x+6} - \sqrt{x^2-5x+4})^{x/2}=2^{\frac{x+4}{4}}$, find $x$.
My method : I first began by substituting $x^2-5x+5$ as $t$. This makes my equation become : $$(\sqrt{t+1} + \sqrt{t-1})^{x/2} + (\sqrt{t+1} - \sqrt{t-1})^{x/2}=2^{\frac{x+4}{4}}$$. I tried squaring but it got even more complicated. Please help me.
Put $A = \left(\sqrt{t+1} + \sqrt{t-1}\right)^{\frac{x}{2}}, B = \left(\sqrt{t+1}-\sqrt{t-1}\right)^{\frac{x}{2}}$, then apply the AM-GM inequality: $A+B \ge 2\sqrt{AB}$ and note that $AB = 2^{\frac{x}{2}}$. Thus the left side $\ge $ the right side, and we have equality so $A = B \implies t = 1 \implies x^2-5x+5 = 1 \implies (x-1)(x-4) = 0 \implies x = 1,4$.