Radicals

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There are 11 real values of x as follows:

{x = 14400, x = 14161, x = 13456, x = 12321, x = 10816, x = 9025, x = 7056, x = 5041, x = 3136, x = 1521, x = 400}

y  = √ [ 120 - √x ]      where  y is an  integer

Note that   120 - √x  must be   ≥  0

One way of solving this is with a graph : https://www.desmos.com/calculator/xmrvja05tc

x values     y  values

14400              0

14161              1

13456              2

12480              3

10816              4

 9025               5

 7056               6

 5041               7

 3136               8

 1521               9

  400              10

So....11 values  of x  produce an  integer  output  for  y

So