Write answer with only one root

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Homework Statement



At the end I get the answer: sqrt(sqrt(3)+2)

It can be show that it is ( sqrt(6)+sqrt(2) ) / 2

How do I show this?, I have tried, to attempt it my manipulating the expression, but I can't seem to manage it.
 
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((sqrt(6)+sqrt(2)) / 2)^2 = (1/4)(6+2*sqrt(2)*sqrt(6) + 2) = (1/4)(8 + 2sqrt(12)) = 1/4(8 + 4sqrt(3)) = sqrt(3) + 2
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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