I have a complex definition of a few variables (that I got using
s = Solve[1 - a x^2 - x^3 == 0, x]) and I now want to know under what conditions of "a" will each of the solutions be real.
For example, one of the variables I have is
r3=-(a/3) + ((1 - I Sqrt) a^2)/(
3 2^(2/3) (-27 + 2 a^3 + 3 Sqrt Sqrt[27 - 4 a^3])^(
1/3)) + ((1 + I Sqrt) (-27 + 2 a^3 +
3 Sqrt Sqrt[27 - 4 a^3])^(1/3))/(6 2^(1/3)), I want to know under what conditions will r3 be a real number.
I have tried simplifying this to the most basic example where I want to know what the conditions on a+b*I need to be so that it is real (where the answer I expect would be b=0), so I tried using
Resolve[a + b I, Reals] but this does not give me the answer I expect.