##### DEFINITION

Completing the square is a mathematical method used to solve quadratic equations. Quadratic equations are those in the following form: ax² + bx + c = d

In this method, a quadratic equation (as above) is usually written as the sum of a perfect square and constant (as below), making it easier to solve: (x + f)² + g = h

The process of finding f and g is given by this method, thus allowing one to find the value of x. Note that x² is considered a square hence x is considered a root**[1]**.

##### INVENTION

The Persian polymath Muhammad Ibn Musa Al-Khwarizmi invented**[2]** the completing the square method around 820. In his publication (circa 820), he demonstrates this method in solving a problem titled, “A square and ten roots are equal to thirty-nine Dirhams”. In the form of a mathematical equation, this is: x² + 10x = 39

It can be taken that a = 1, b = 10, c = 0, d = 39, as per the definition. This must be treated geometrically, as below. It is implied that a is factored out (since a = 1 here, there is no observable change):