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Visualising the complex roots of quadratic equations with real coefficients. (English)
Aust. Sr. Math. J. 26, No. 2, 6-20 (2012).
Summary: The roots of the general quadratic equation $y = ax^2 + bx + c$ ($a$, $b$, $c \in \Bbb R$) are known to occur in the following sets: (i) real and distinct; (ii) real and coincident; and (iii) a complex conjugate pair. Case (iii), which provides the focus for this investigation, can only occur when the values of the real coefficients $a$, $b$, and $c$ are such as to render the discriminant negative. In this case, a simple two-dimensional $x$-$y$ plot of the quadratic equation does not reveal the location of the complex conjugate roots, and the interested student might well be forgiven for asking, “Where exactly are the roots located and why can’t I see them?" In the author’s experience, this sort of question is hardly ever raised ‒ or answered satisfactorily ‒ in school years 11 or 12, or in undergraduate mathematics courses. In this paper, the author aims to provide a clear answer to this question by revealing the whereabouts of the complex roots and explaining the significance of the conjugate pairing. (ERIC)
Classification: H34 I24 F54