This applet demonstrates a geometric interpretation of the method of completing the square. The construction was first described during the 9-th century by the great Persian mathematician Al-Khwarizmi in his book [i]The Compendious Book on Calculation by Completion and Balancing[/i].[br]We consider the equation [math]x^2+bx=c[/math].[br] In Al-Khwarizmi’s interpretation [math]x[/math] and [math]b[/math] are the lengths of the sides of a square ([math]x[/math] by [math]x[/math]) and a rectangle ([math]b[/math] by [math]x[/math]), and [math]c[/math] is the sum of the areas of the two, therefore all three values must be positive. Al-Khwarizmi used the construction below to find the positive solution of the equation. We will use Al-Khwarizmi’s geometric reasoning to complete the square, but then finish the solution and find possible negative values by using the methods of symbolic algebra.

The following construction (not described by Al-Khwarizmi) is based on an equation of the form [math]x^2-bx=c[/math]. Since [math]x[/math] and [math]b[/math] must be positive, then for [math]c[/math] to be positive, [math]x[/math] must be greater than [math]b[/math]. [br][br]