The discriminant is the part of the quadratic formula under the radical, namely:
Two real and unequal roots. If is a perfect square, the roots are rational. Otherwise, they are irrational.
One real root with a multiplicity of two. That is to say that the trinomial is a perfect square and has two identical factors. The Fundamental Theorem of Algebra still holds because it allows counting roots up to the limits of their multiplicity.
A conjugate pair of complex roots of the form where is the imaginary number defined by
Terminology note: Rarely will you find a quadratic with purely imaginary roots. A quadratic that does not have real roots generally has complex number solutions which have a real part and an imaginary part. Hence, to say that a quadratic has "two different imaginary solutions" is almost always incorrect.
