Methods for solving quadratic equations
Dave L. Renfro
discussion group and thought that it would
be useful to also archive it in sci.math,
where a google usenet search would bring
it up.
Dave L. Renfro wrote:
http://mathforum.org/kb/message.jspa?messageID=4780597
> You could assemble brief outlines of 10-15 methods
> for solving quadratic equations and have students,
> in groups of 2-3, prepare a presentation to the
> class on the methods (one method per group). You
> can find many unusual methods in history of math
> books at a university library, for example. If
> you'd like, I can post some journal references,
> but let me know, because I'll need to look them
> up when I get home tonight if you want them.
I didn't see a reply by the original poster to
my offer, but I thought it would be a good idea
to archive what I had in mind in a math-teach post.
Not all of this would be appropriate for the
original poster's students, but all of it should
be accessible and relevant for a high school
teacher (even the slide rule method, if only
for historical enrichment purposes).
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Caroline Eucebia Schuler [Shuler], "Application of
professional treatment to the quadratic function",
School Science and Mathematics 37 #5 (May 1937), 536-548.
William L. Schaaf, "Some reflections on quadratic
equations", Memorabilia Mathematica column, Mathematics
Teacher 49 #8 (December 1956), 618-621.
Schuler gives a nice survey on the history of,
and on various ways of solving, quadratic equations.
Her bibliography contains 32 items.
Schaaf Gives 47 references dealing with quadratic
equations from elementary level journals.
For more recent references, a good place to begin
is the first article I gave in the following May 10,
2006 math-history-list post (see also the three
follow-up posts by others dated May 11):
http://mathforum.org/kb/message.jspa?messageID=4702130
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What follows is part of the Mathematical Miscellanea
column edited by Phillip S. Jones in Mathematics
Teacher 44 #3 (March 1951), pp. 193-194 (pages
for this part only, not the entire column). In
what follows, italicized words and phrases are
enclosed in single quote marks (') and usual
double quote marks (") are used where the
original uses them.
A letter from Willian [sic?] J. Hazard of the
Department of Engineering Mathematics of the
University of Colorado includes the following
list of 18 ways to solve ax^2 + bx + c = 0 taken
from an article which he published in January 1924
in the 'Colorado Engineer':
1. By factoring by inspection.
2. By factoring after a substitution, z = ax,
which leads to z^2 + bz + ac = 0.
3. By factoring in pairs by splitting bx into
two terms.
4. By completing the square when a is 1
and b is even.
5. By completing the square as usual after
dividing through by a.
6. By completing the square by the Hindu
method ("the pulverizer"), i.e. by multiplying
through by 4a and adding b^2 to both sides.
7. By completing square as given, adding b^2/4a.
8. By the formula.
9. By trigonometric methods (see Wentworth-Smith,
'Plane Trigonometry').
10. By slide rule (see Joseph Lipka, 'Graphical
and Mechanical Computation'. John Wiley and Sons,
Inc. [1918], p. 11 ff.
11. By graphing for real roots. (All modern
textbooks.)
12. By graph, extended for complex roots.
(See: Howard F. Fehr, "Graphical Representation
of Complex Roots," 'Multi-Sensory Aids in the
Teaching of Mathematics', 'Eighteenth Yearbook
of the National Council of Teachers of Mathematics'
[1945] pp. 130-138. George A. Yanosik, "Graphical
Solutions for Complex Roots of Quadratics, Cubics,
and Quartics," 'National Mathematics Magazine', 17
[Jan. 1943], pp. 147-150.)
13. Real roots by Lill circle. (d'Ocagne, 'Calcul
graphique et nomographie', from which L. E.
Dickson got his reference to it in his 'Elementary
Theory of Equations'.) (Also see J. W. A. Young's
'Monographs on, Topics in Modern Mathematics'
"Constructions with Ruler and Compasses.")
14. By extension of the Lill circle to include
complex roots.
15. Using the graph of y = x^2 and y = -bx - c
to find real roots. (Lipka, 'op. cit.' p. 26,
modifies and extends this solution; Schultze,
'Graphic Algebra'; Hamilton and Kettle, 'Graphs
and Imaginaries'.).
16. By extending (15) to include complex roots
(Hamilton and Kettle, Schultze).
17. By use of a table of quarter squares. This
is a practical method of handling an equation
having large constants, as we already have the
table in print (Jones' 'Mathematical Tables').
18. By use of "Form Factors."
Professor Hazard adds that methods 12, 14, 17, 18
are original with himself, and that 13, 14 and 17
will be discussed in his book 'Algebra Notes' to
be published soon. He also suggests a 19th method:
"Make a template of transparent plastic, very
slightly smaller and parallel to the unit parabola
y = x^2, so that the pencil or ruling pen will
draw the true curve. Scratch co-ordinates on it
for accurate placing, and for any equation move
the template to a position parallel to the Y axis
and with its vertex at the point
( -b/2a , (4ac - b^2)/4a ).
While this can't be called a different 'method'
of solution, I think it should be included as a
very handy and practical 'variation of approach'
to the graphic solution of the quadratic in
one unknown."
We hope that Professor Hazard and others will
extend our list and include expositions of
some of the less obvious or less easily
available procedures.
Before leaving this topic, however, we should
mention the reprint sent us by Professor
H. D. Larsen of "Solutions of the Quadratic
Equation" which appeared in the Spring 1950
issue of 'The Pentagon'. Professor Larsen,
editor of the journal, writes that he has
some copies of the article which he will
sell for thirty-five cents each. The article
was written by Raymond H. Gillespie as an
Honors Thesis in Mathematics while he was
an undergraduate at Albion College, Albion,
Michigan where Professor Larsen is also
Chairman of the Mathematics Department.
This thesis includes an historical survey,
early solutions (including Viete's solution
via the substitution of x = u + z in
x^2 + ax + b = 0 and the equating of the
coefficient of u to 0), derivations of the
formula (including that via determinants
by Euler and Bezout), methods of factoring,
graphical solutions, determination of
imaginary roots, and solutions by trigonometric
methods. Although the tabulation of procedures
is not, of course, absolutely complete and
the historical survey ignores entirely recent
discoveries in regard to Babylonian achievements,
nevertheless it is well done and contains
much interesting material which would enrich
both classroom teaching and club programs.
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Dave L. Renfro