<s>[INST] How does $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ simplify to $1 - \sqrt 2 $? I've the answer for a question in my textbook to be:
$-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ which i've then simplifed to: $-\sqrt {3 - 2\sqrt 2 } $
However my textbook states $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} =  1 - \sqrt 2 $
How is this?
 [/INST] Alternatively, we can say $x = -\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}$. We can say that $x^2 = -\frac{2-\sqrt{2}}{2+\sqrt{2}} \implies -(2+\sqrt{2})x^2 + (-2 + \sqrt{2}) = 0$. We can now apply the quadratic formula and discard the negative root to obtain that $x = -\frac{2\sqrt{2}}{4+2\sqrt{2}} = -\frac{\sqrt{2}}{2+\sqrt{2}} = -\frac{2}{2+2\sqrt{2}} = -\frac{1}{1+\sqrt{2}} = 1-\sqrt{2}.$ 
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