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Approaching quadratic equations from a right angle

Published online by Cambridge University Press:  16 October 2017

King-Shun Leung*
Affiliation:
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, Hong Kong e-mail: [email protected]

Extract

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

Type
Articles
Copyright
Copyright © Mathematical Association 2017 

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References

1. Allaire, Patricia R. and Bradley, Robert E., Geometric approaches to quadratic equations from other times and places, Mathematics Teacher, 94 (4) (April 2001) pp. 308313.Google Scholar
2. Bradford, Phillips Verner, Visualizing solutions to n-th degree algebraic equations using right-angle geometric paths - Extending Lill's method of 1867, accessed March 2017 at: http://web.archive.org/web/20100502013959/ http://www.concentric.net/-pvb/ALG/rightpaths.html Google Scholar
3. Martin, George E., Geometric Constructions, Springer-Verlag, New York (1998).Google Scholar
4. Yates, Robert C., The angle ruler, the marked ruler and the carpenter's square, National Mathematics Magazine, 15 (2) (November 1940) pp. 6173.CrossRefGoogle Scholar