+288 I bet many people are falling for this:
sqrt(x^2) does not necessarily equal x.
I repeat, sqrt(x^2) does NOT Necessasrliy equal x.
What if x is negative? :)
We'll have to take 2 cases:
1. x >= 0;
This gives us sqrt(120 - x) is an integer.
This means that 120-x is a perfect square. Since x is nonnegative, the amount of values of x is the amount of perfect squares less than 120.
This ranges from 1^2.... to 10^2.. Therefore there are 10 values where x is nonnegative.
2. x < 0
This gives us that sqrt(120 + x) is an integer. However, note that x is negative and there will be the same values of x before because x in our first case is like -x in our case here. Hence, there are 10 values where x is negative.
Adding up these two cases, gives us 20 values.
