For a real number x, find the number of different possible values of x^2 - abs(x)^2.

one.

x^2 - l x^2 l

Note that the absolute value of x^2 = x^2

So......this always evaluates to 0

Yes you are very right.

For anyone reading this solution, x has to be a real number as declared in the question, thus x^2 - |x^2| could not be negative...

x can't be i.