Question 245864
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The discriminant is the part of the quadratic formula under the radical, namely:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \Delta\ =\ b^2\ -\ 4ac]


*[tex \LARGE \Delta > 0 \ \ \Rightarrow\ \] Two real and unequal roots.  If *[tex \LARGE \Delta] is a perfect square, the roots are rational.  Otherwise, they are irrational.


*[tex \LARGE \Delta = 0 \ \ \Rightarrow\ \] 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.


*[tex \LARGE \Delta < 0 \ \ \Rightarrow\ \] A conjugate pair of complex roots of the form *[tex \LARGE a \pm bi] where *[tex \LARGE i] is the imaginary number defined by *[tex \LARGE i^2 = -1]


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.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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