Find the value
\[ \sqrt{5 + 2\sqrt{6}} \]
(a) \(\sqrt{3} – \sqrt{2}\) \quad (b) \(\sqrt{3} + \sqrt{2}\) \quad (c) \(\sqrt{5} + \sqrt{6}\) \quad (d) none of these
Solution:
\[ \sqrt{5 + 2\sqrt{6}} = \sqrt{a} + \sqrt{b} \]
\[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \]
\[ a + b = 5,\quad ab = 6 \Rightarrow a = 3,\ b = 2 \]
\[ \therefore \sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2} \]
\[ \therefore \text{Correct Answer: } \sqrt{3} + \sqrt{2} \ (\text{Option b}) \]