When is the discriminant of a quadratic function imaginary?
The discriminant of a quadratic function can only be imaginary if at least some of the coefficients of the quadratic are imaginary.
For a quadratic in the general form
The discriminant is
If the discriminant is negative (which might be what you intended to ask)
the square root of the discriminant is imaginary
and therefore the quadratic formula
gives imaginary values as roots for
This happens when the parabola does not touch or cross the X-axis.