id: 06186010
dt: j
an: 2013d.00569
au: Bardell, Nicholas S.
ti: Visualising the complex roots of quadratic equations with real
coefficients.
so: Aust. Sr. Math. J. 26, No. 2, 6-20 (2012).
py: 2012
pu: Australian Association of Mathematics Teachers (AAMT), Adelaide, SA
la: EN
cc: H34 I24 F54
ut: algebra; equations; mathematics curriculum; high school students; quadratic
functions; roots; complex numbers
ci:
li: http://www.aamt.edu.au/Webshop/Entire-catalogue/Australian-Senior-Mathematics-Journal
ab: 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)
rv: