Strictly speaking, from the way you worded your question, there are no real values of for which has purely imaginary roots. That is because the coefficient on the 1st degree term is non-zero and rational, hence any roots of the given equation must have a rational (and therefore real) component. There is a range of values for for which will have a pair of complex roots of the form where and .
Remember the discriminant:
If then the roots are a conjugate pair of complex numbers of the form . Only in the case where is the real part of the complex number equal to zero and the complex number purely imaginary.
So:
And for , , hence or .
Actually it is possible to have purely imaginary roots if you allow , but somehow I don't think that is what you were talking about when you asked the question.
