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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[3]) a^2)/( 3 2^(2/3) (-27 + 2 a^3 + 3 Sqrt[3] Sqrt[27 - 4 a^3])^( 1/3)) + ((1 + I Sqrt[3]) (-27 + 2 a^3 + 3 Sqrt[3] 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.

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You can just add a domain specification to Solve or NSolve:

NSolve[1 - a x^2 - x^3 == 0, x, Reals]

{{x -> ConditionalExpression[Root[-1. + a #1^2 + #1^3 &, 1], a > 1.88988 || a < 1.88988]}, {x -> ConditionalExpression[Root[-1. + a #1^2 + #1^3 &, 2], a > 1.88988]}, {x -> ConditionalExpression[Root[-1. + a #1^2 + #1^3 &, 3], a > 1.88988]}}

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