Since we are asked for x+y, it suggests that even if there are several possible values for x and y, that x+y can take on only one value.  So assume x=y.

If x=y then x + √x - √17 = 0  quadratic in √x

And with x=y, x+y = 2x

Domain of x:  x>0

And plugging in x=2 and x=3 shows 2 < x < 3.

So 4 < (x+y) < 6

Quadratic equation to solve for √x:

√x = (-1 ± sqrt(1 + 4√17))/2

x = (1 + (1 + 4√17) ± 2sqrt(1 + 4√17))/4

x = (1 + 2√17 ± sqrt(1 + 4√17))/2

The minus version is about 5.    the plus version is about 13.4

So reject the plus version.

x = (1 + 2√17 - sqrt(1 + 4√17))/2

(x+y) = 1 + 2√17 - sqrt(1 + 4√17)

about 5.0638    which agrees with broll's solution.