[size=150] This applet demonstrates a geometric interpretation of the method of completing the square. The construction was first described during the 9[sup]th[/sup] 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].[/size][br]

1. Slowly move the slider to see [br][list][*]an x-by-x square with area x[sup]2 [/sup][br][/*][*]a 10-by-x rectangle with area 10x.[/*][/list]The blue area is therefore: x[sup]2[/sup]+10x. [br][br]2. Keep slowly sliding. Next we divide the rectangle into two parts, and move the parts, getting ready to form a SQUARE.[br][br]3. Keep slowly sliding. We are COMPLETING THE SQUARE (geometrically, instead of algebraically.) The orange square completes our large square. [br][br]

4. Thus, to [b]COMPLETE THE SQUARE[/b] for x[sup]2[/sup]+10x, we add 5[sup]2[/sup] to get:[br][br] [b] x[sup]2[/sup]+10x+25[/b]