Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers

Daniel E. Otero (Xavier University)


Nearly every student of mathematics, after receiving years of training in the rules and procedures of arithmetic, enters the realm of "higher" mathematics through the study of algebraic problem solving, the finding of unknown quantities from known arithmetical conditions on those quantities. Algebra is a staple of the secondary school curriculum around the world, and a standard rite of passage for students in this curriculum is some form of mastery of the process of factoring polynomials, especially quadratic polynomials. Learning to factor quadratics is a precursor to a complete treatment of quadratic equations and their solutions, including the procedure known as "completing the square" and culminating with the well-known quadratic formula: given a quadratic equation of the form \(ax^2+bx+c=0\) (with \(a\neq0\)), its solutions are given by \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] The mini-Primary Source Project (mini-PSP) Completing the Square: From the Roots of Algebra presented here is designed to give students a deep understanding of the method of completing the square, which serves as a bridge between the method of factoring and the quadratic formula.

Beginning algebra students naturally focus their attention on mastering the procedures of equation solving and learning to get the correct answers, ignoring questions like "Why and how do the procedures work?" For them, completing the square can involve procedural steps that mysteriously produce the required sought-for answers, and the quadratic formula can act like a runic talisman that magically generates the right numbers that solve the given equation. And even if the question "Why?" is seriously considered in this context, many textbooks answer with a symbolic derivation that is complicated and unsatisfying. The best answer to this question is quite naturally found in the history of the development of this method.

Although problems of quadratic type have been posed and solved for thousands of years, the systematic approach to algebraic problem-solving goes back to the "Father of Algebra," Muḥammad ibn Mūsā al-Khwārizmī (ca 780–850 CE), a ninth-century scholar who wrote in Arabic in the then-young city of Baghdad under the patronage of one of the great caliphs of the Islamic Abbasid Empire. Written in about the year 825, al-Khwārizmī 's extremely influential work on the subject, with the title al-Kitāb al-mukhtaar fī hisāb al-jabr wal-muqābala (The Compendious Book on Calculation by Restoration and Reduction), better known today simply as Algebra, instructs his readers how to find the roots of an equation. But al-Khwārizmī's equations are ones without symbols; they are expressed entirely in words. This rhetorical algebra of al-Khwārizmī provides a careful description of the method we call completing the square, along with a clear geometric demonstration of how the method works that involves completing a real (geometric) square!

A page from an Arabic copy of al-Khwārizmī's Algebra,

In the project Completing the Square: From the Roots of Algebra, students work through selections from al-Khwārizmī's Algebra, using text from two English translations of the work [Rāshid 2009, Rosen 1831]. In a pair of appendices, students can then further explore these ideas through (1) a derivation of the quadratic formula, and/or (2) consideration of when a quadratic equation produces complex roots. The project is meant to serve multiple needs: it can be used by students who are learning algebraic methods for solving quadratic equations for the first time; by future high school mathematics teachers who will be responsible for teaching algebra in their own classrooms; and by students in a general history of mathematics course as an introduction to the role of early Islamic-era mathematics in the development of algebra.

The complete project Completing the Square: From the Roots of Algebra (pdf)  is ready for student use, and the LaTeX source code is available from the author by request. A set of instructor notes that explain the purpose of the project and guide the instructor through the goals of each of the individual sections is appended at the end of the student project.

This project is the twelfth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources appearing in Convergence, for use in courses ranging from first year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection also offers dozens of other mini-PSPs and a similar number of more extensive full-length PSPs which are meant for other topics across the undergraduate mathematics curriculum, including a longer version of this project (entitled Solving Equations and Completing the Square: From the Roots of Algebra) that includes an introduction to the rhetorical algebra of al-Khwārizmī and a deeper exploration of his solutions to quadratic equations.


The development of the student project Completing the Square: From the Roots of Algebra has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.


Katz, Victor, Imhausen, Annette, Robson, Eleanor, Dauben, Joseph W., Plofker, Kim, and J. Lennart Berggren, editors. 2007. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton NJ: Princeton University Press.

Rāshid, Rushdī. 2009. Al-Khwārizmī: The Beginnings of Algebra. London: Saqi.

Rosen, Frederic. 1831. The Algebra of Moḥammed ben Mūsā, Translated and Edited by Frederic Rosen. London: Oriental Translation Fund.