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Completing the Square

Author:
Irina Boyadzhiev
Topic:
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.
GeoGebra Applet
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 .
GeoGebra Applet

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