Find the value of \(A\)
\[ \sqrt{5 + 2\sqrt{6}} = A + \sqrt{3} \]
Solution:
\[ (\,A + \sqrt{3}\,)^2 = 5 + 2\sqrt{6} \]
\[ A^2 + 3 + 2A\sqrt{3} = 5 + 2\sqrt{6} \]
Comparing rational and irrational parts:
\[ A^2 + 3 = 5 \Rightarrow A^2 = 2 \]
\[ 2A\sqrt{3} = 2\sqrt{6} \Rightarrow A\sqrt{3} = \sqrt{6} \Rightarrow A = \sqrt{2} \]
\[ \therefore A = \sqrt{2} \]