If $$\frac{5 + 2\sqrt3}{2-\sqrt3} = x + y\sqrt3,$$ then (x, y) = ?

Correct option is D

Given:

​$$\frac{5 + 2\sqrt3}{2-\sqrt3} = x + y\sqrt3$$​​

Solution:

$$\frac{5 + 2\sqrt3}{2-\sqrt3} = x + y\sqrt3$$ 

Rationalizing the LHS side

​$$\frac{5 + 2\sqrt3}{2-\sqrt3}\times \frac{2+\sqrt3}{2+\sqrt3} = x + y\sqrt3$$​​

​$$\frac{(5 + 2\sqrt3)(2+ \sqrt{3})}{(2)^-(\sqrt3)^2} = x + y\sqrt3$$​​

​$$\frac{10+5\sqrt{3} +4\sqrt{3} + 2(3) }{4-3} = x + y\sqrt3$$​​

​$$16+9\sqrt{3}= x + y\sqrt3$$​​

Comparing LHS and RHS

We get x =16 and y = 9

Hence (x,y) = (16,9)