Question 199156
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Each of these problems is in the form:


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


For each of them, calculate the discriminant (*[tex \Large \Delta]):


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


Then evaluate the character of the roots based on the value of *[tex \Large \Delta] according to the following criteria (which presume rational coefficients on your quadratic):


*[tex \LARGE \Delta > 0 \ \ \Rightarrow\ \] Two real and unequal roots.  If *[tex \Large \Delta] is a perfect square, then both roots are rational.  Otherwise the two roots are a conjugate pair of irrational roots of the form *[tex \LARGE a \pm b] where *[tex \LARGE a] is rational and *[tex \LARGE b] is 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.


*[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]
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