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Completing the Square

Barnabas Hughes

A major goal for secondary school students in their study of elementary algebra is to understand, solve, and apply the quadratic equation.  Modern texts commonly begin with an application of the quadratic equation focused on the parabola.  This is a grand improvement over the text used by the author that began with the bald statement, “Quadratic Equations.  An equation of the second degree is called a quadratic equation.”  Immediately the student was plunged into the zero law (\( ab = 0 ,\) etc.), and how to solve quadratic equations by factoring.  Only after struggling through 73 exercises would the students be challenged with some practical applications.  Useful as is factoring, it is not the original way of solving quadratic equations.

The quadratic equation, as we know it today, was first discussed and taught by Muhammed ibn Musa al-Khwarizmi (fl. 815-850).  At the command of his Caliph, he collected all the material he could find on algebra and wrote the first text on the subject.  The title of his book contains the word algebra.  After brief attention to first degree equations and simple quadratics that required only square roots for their solution, he turned to quadratic equations.  While he did not use the word equation, the quadratic equation is correctly named:  it focuses on the dimensions of a square.  In fact, quadratic means square.  Al-Khwarizmi, as Muhammed is more commonly called, solved quadratic equations by the method we call today, completing the square.  Again, this phrase describes exactly what he did, as seen in the solution of his example that is the classical quadratic equation, \[ x^2 + 10x = 39 .\]

In the upper left section of the illustration below, the terms of the equation, \( x^2 \) and \( 10x ,\) are represented by geometric figures, a square and a rectangle.  The sum of their areas is given to be \( 39 .\)  In the lower left section of the illustration, the rectangle is cut into two parts that are attached to adjacent sides of the square.  The L-shaped result is then filled in with a smaller square that fills out or completes the larger square.  The addition increases the given size of the figure from \( 39\) to \( 39 + 25 = 64 .\)  The solution follows easily.



Exercises:  (1) \( x^2 + 12x = 108 ;\)  (2) \( x^2 + 6x = 55 ;\)  (3) \( x^2 + 8x = 65 ;\)  (4) \( x^2 + 4x = 60 .\)

Barnabas Hughes, "Completing the Square," Loci (August 2011)