
06186010
j
2013d.00569
Bardell, Nicholas S.
Visualising the complex roots of quadratic equations with real coefficients.
Aust. Sr. Math. J. 26, No. 2, 620 (2012).
2012
Australian Association of Mathematics Teachers (AAMT), Adelaide, SA
EN
H34
I24
F54
algebra
equations
mathematics curriculum
high school students
quadratic functions
roots
complex numbers
http://www.aamt.edu.au/Webshop/Entirecatalogue/AustralianSeniorMathematicsJournal
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 twodimensional $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)