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Completing the Square

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 The Compendious Book on Calculation by Completion and Balancing. We consider the equation . In Al-Khwarizmi’s interpretation  and are the lengths of the sides of a square ( by ) and a rectangle ( by ), and 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 . Since and must be positive, then for to be positive, must be greater than .