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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.

The article examines geometrical problems encountered in locating or determining the normals to a parabola and the real roots of a cubic. In the book "Tractatus duo, prior de cycloide, posterior de cissoide," John Wallis made the first rectification of a algebraic curve. The transformation of any cubic equation through the horizontal translation do not alter the number of real roots of the equation. Key information about the Neil's parabola and envelope of the family of normals is further presented.