Quadratic Equation Solution - Completing the Square

Al-Khawarizmi was a ninth-century mathematician who developed the solution to the quadratic equation we commonly refer to as completing the square. It is an appealing visual, geometric proof. 

Example: Given the following quadratic equation, solve for the value x.

This equation can be represented by a sum of a square with side x and a rectangle wtith sides of 8 and x.

Now split the rectangle into two equal halves forming rectangles with sides of 4 and x. Rearrange the rectangles around the square. The sum of the areas remain the same.

Complete the square - make the full arrangement square by adding a smaller square in the 4 x 4 gap (the yellow square below). The new area of the total arrangement is 65 + 16 = 81. The sides of the larger square are 9, the square root of 81.

Since each side of the completed square is 9, x = 9 - 4 = 5. Confirm the solution by

 substituting for x in the original equation:

5 x 5 + 8 x 5 = 65