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The geometric problem of finding the number of normals to the parabola y = x[squared] through a given point is equivalent to the algebraic problem of finding the number of distinct real roots of a cubic equation. Apollonius solved the former problem, and Cardano gave a solution to the latter. The two problems are bridged by Neil's (semi-cubical) parabola.

Many students, when they take an elementary differential equations course for the first time, bring with them misconceptions from numerical methods that they had learnt in their calculus courses, most notable of which concerns the mesh width in using a numerical method. It is important that we strive to dispel any of these misconceptions as well as to make students aware that there is often a trade-off between the complexity of a numerical method and its computational cost. In this note, we provide examples to do that. (Contains 4 footnotes, 2 tables, and 7 figures.)

Mathematics is a human endeavor, and the story of mathematics is part of the larger story of humankind. There are many opportunities for us to use the history of mathematics, either as a vehicle for presenting topics or as a sidebar, to draw our students into the larger story of mathematics and, thereby, increase their interest in our subject. It works because, after all, there is nothing like a good story to get one's attention. We present one example from the history of mathematics, Piflgala's method, that can be used as a sidebar in a variety of courses, from college arithmetic to elementary or intermediate algebra to linear algebra. The problem is simple: How can we find an if a is a real or complex number without having to find n - 1 products (or, worse, An if A is a square matrix)? [ABSTRACT FROM AUTHOR]

The article reports on the cubic equation's number of real roots. To illustrate, it mentions about the number of normals to the parabola by a given point in the plane. It states that to find the number of real roots, Cardan's cubic formula is useful, but not easily in general. To tell the real root's number without firts solving the equation, discrimination is suggested. It presents how the discriminant defines a boundary in the plane. These ideas are showed from a first calculus course: critical point, derivative, and graphing in an intuitive way.