Why is a Square Square and a Cube Cubical?

Summary

Introduction

Why is the algebraic process of forming a perfect quadratic expression referred to as completing the square? With simple cutouts for an overhead projector, the geometric underpinnings as well as many aspects of quadratic equations can be exhibited quickly and effectively. This can be expanded to the cubic with the use of five wooden blocks. This chapter will show how to use these items in the classroom to give students a geometrically intuitive as well as an historical understanding of quadratic and cubic expressions.

Creating squares

At several points in the undergraduate curriculum, as well as in some secondary courses, we find ourselves “completing the square.” I found myself in just such a position a short while back while teaching a sophomore level calculus course. On the spur of the moment, I asked the class if they knew why it was called “completing the square”. The response ranged from blank stares to shrugged shoulders. Looking at the clock, I decided I could sidetrack for a few minutes.

I started by drawing the Figure 1 on the board to represent the statement x² + bx. (In general, you will have some expression of the form ax² + bx + c. But the actual process of completing the square is done on an expression of the form x² + bx, after division by a and “removal” of c.) I start with the comment that for the ancient Greeks as well as the Babylonians, geometry was physical; that x (in modern notation) represented a length, x² an area, and so on.