Question 382465
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For any quadratic polynomial equation of the form:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ +\ c\ = 0]


Find the Discriminant, *[tex \LARGE \Delta\ =\ b^2\ -\ 4ac] and evaluate the nature of the roots as follows:


No calculation quick look:  If the signs on *[tex \Large a] and *[tex \Large c] are opposite, then *[tex \LARGE \Delta > 0] guaranteed.


*[tex \LARGE \Delta > 0 \ \ \Rightarrow\ \] Two real and unequal roots. If *[tex \LARGE \Delta] is a perfect square, the quadratic factors over *[tex \LARGE \mathbb{Q}].


*[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.


*[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]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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