Minimizing a quite complex algebraic expression

Question

I am a junior high school student. I am solving a problem that requires me to determine the minimum of the following two complex algebraic expressions. $$\dfrac{\sqrt{5-4\sqrt{-x^2+2\sqrt{3}x-2}}+2\sqrt{2\sqrt{3}x+2\sqrt{-x^2+2\sqrt{3}x-2}-1}}{2}$$ $$\dfrac{\sqrt{5+4\sqrt{-x^2+2\sqrt{3}x-2}}+2\sqrt{2\sqrt{3}x-2\sqrt{-{x}^2+2\sqrt{3}x-2}-1}}{2}$$ $$(\sqrt{3}-1<x<\sqrt{3}+1)$$ However, I am having some difficulty.
Taking the first equation as an example, obviously I need to minimize $5-4\sqrt{-x^2+2\sqrt{3}x-2}$ and $2\sqrt{3}x+2\sqrt{-x^2+2\sqrt{3}x-2}-1$.
This creates a contradiction: on the one hand I want to maximize $-x^2+2\sqrt{3}x-2$, and on the other hand I want to minimize $-x^2+2\sqrt{3}x-2$ (which actually has no minimum), making it impossible for me to successfully minimize this expression.
WolframAlpha told me the minimum is $\dfrac{\sqrt{5}}{2}+\sqrt{5+2\sqrt{3}}$ when $x$ takes on an extreme value of the interval $\sqrt{3}-1$, but I don't know how it works out.(Step-by-step solution is unavaliable)
I would be very grateful if someone could give me some help.

Answer 1

This looks a little too complicated for junior high school student but anyway here are some hints:

• $-x^{2}+2\sqrt{3}x-2=1-\left(x-\sqrt{3}\right)^{2}$
• $x=\sqrt{3}+\cos{\left(\theta\right)}\phantom{x},\phantom{x}\theta\in\left(0,\pi\right)$

Substitute these into the equation to obtain the following expressions:

$$ \frac{\sqrt{5-4\sin{\left(\theta\right)}}+2\sqrt{5+4\sin{\left(\theta+\frac{\pi}{3}\right)}}}{2} $$

$$ \frac{\sqrt{5+4\sin{\left(\theta\right)}}+2\sqrt{5-4\sin{\left(\theta-\frac{\pi}{3}\right)}}}{2} $$