Question 198819
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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]


By the way, if you thought that you could just drop your entire homework assignment in here and get someone to do it for you, let me disabuse you of that  notion right now.


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