Formula

 Completing the square is a fun little exercise, but it is also crucial

to master. It is, in effect, the royal road to the quadratic formula,

the infaillible algorithm for solving all quadratics. Below, how it is

presented in Math is Fun.

Mathematicians have been using it since Antiquity, but using

words instead of mathematical expressions. Indeed, it was only 

with Descartes in the 17th Century that we got down to using the

plan affine, letters for unknows, everything on the left and equal to zero.

And yet to come was a systematic appreciations of the roots question: 

two roots with understandable exceptions.

Oh wait... If we decompose it, -b/2a appears as the first element. Our

point of inflection. Then plus or minus one and the same value...

                                                         *     *     *

The quadratic formula:

                                                             

source: Wikipedia

*     *     *

Math is Fun:

                                                                               

                                                                                 

*     *     *

Al-Khwarizmi

                                                                                

                                                                                    

                                                                       *     *     *

An intuitive proof:

                                                                       

        source: Russian-language Wikipedia