title: Mathematica FullSimplify[Sqrt[5+2 Sqrt[6]]] yields Sqrt[2]+Sqrt[3] but FullSimplify[-Sqrt[5+2 Sqrt[6]]] is not simplified, why?
body: <p>I was playing with the (beautiful) polynomial <code>x^4 - 10x^2 + 1</code>.
  Look what happens:</p>
  
  <pre><code> In[46]:= f[x_] := x^4 - 10x^2 + 1
            a = Sqrt[2];
            b = Sqrt[3];
            Simplify[f[ a + b]]
            Simplify[f[ a - b]]
            Simplify[f[-a + b]]
            Simplify[f[-a - b]]
   Out[49]= 0
   Out[50]= 0
   Out[51]= 0
   Out[52]= 0
  
   In[53]:= Solve[f[x] == 0, x]
   Out[53]= {{x-&gt;-Sqrt[5-2 Sqrt[6]]},{x-&gt;Sqrt[5-2 Sqrt[6]]},{x-&gt;-Sqrt[5+2 Sqrt[6]]},{x-&gt;Sqrt[5+2 Sqrt[6]]}}
   In[54]:= Simplify[Solve[f[x] == 0, x]]
   Out[54]= {{x-&gt;-Sqrt[5-2 Sqrt[6]]},{x-&gt;Sqrt[5-2 Sqrt[6]]},{x-&gt;-Sqrt[5+2 Sqrt[6]]},{x-&gt;Sqrt[5+2 Sqrt[6]]}}
   In[55]:= FullSimplify[Solve[f[x] == 0, x]]
   Out[55]= {{x-&gt;Sqrt[2]-Sqrt[3]},{x-&gt;Sqrt[5-2 Sqrt[6]]},{x-&gt;-Sqrt[5+2 Sqrt[6]]},{x-&gt;Sqrt[2]+Sqrt[3]}}
  </code></pre>
  
  <p><code>Sqrt[5-2 Sqrt[6]]</code> is equal to <code>Sqrt[3]-Sqrt[2]</code>.<br>
  However, Mathematica's <code>FullSimplify</code> does not simplify <code>Sqrt[5-2 Sqrt[6]]</code>.</p>
  
  <p><strong>Question: Should I use other more specialized functions to algebraically solve the equation? If so, which one?</strong></p>