What is the sum of x+y?

Question

Can someone please help me in solving this problem.

if $\sqrt{x} + \sqrt{y} = 5+2\sqrt{6}$, then $x + y$ is equal to

(a) $2\sqrt{6}$

(b) $5-2\sqrt{6}$

(c) $5$

(d) $2\sqrt{6}$

Answer 1

If we trust in the strict equality then

$$\sqrt{x} + \sqrt{y} = 5 + 2\sqrt{6}$$

might mean

$$\sqrt{x} = 5 \to x = 25$$

$$\sqrt{y} = 2\sqrt{6}\to y = 24$$

Hence

$$x + y = 25 + 24 = 49 = \text{none of them}$$

Otherwise, just write the equation

$$\sqrt{x} + \sqrt{y} = 5 + 2\sqrt{6}$$

Square it and try to get a solution, which be, by the way, not unique since one parameter is free (either $x$ or $y$).