Question 199506
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*[tex \LARGE \Delta > 0 \ \ \Rightarrow\ \] Two real and unequal roots.


*[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]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2 + 1 = 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x = \frac{-0 \pm sqrt{0^2 - 4(1)(1)}}{2(1)}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \Delta = -4]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x = \frac{ \pm 2i}{2}= \pm i]


Say you want to build a box to store your note cards.  All you know is that your note cards are 2 inches longer than they are wide and the area of the cards is 15 square inches.  What is the minimum dimensions for the inside of the box if you want to store no more than a 4 inch tall stack of note cards?



Let *[tex \Large x] represent the length.  Then *[tex \Large x - 2] is the width.  The area is the length times the width, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x(x - 2) = 15]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 2x - 15 = 0]



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