[wrycoder]

And the third pixelated equation has an error - the final term on the left side should be C, not Cx^2.

I think the point of the arXiv article is this is a more straightforward /proof/ of the quadratic formula. That article has some interesting historical commentary that shows exactly where the author thinks his contribution fits - he is not naively coming up with something "new".

As far as computation of the roots goes, it is a slightly streamlined approach:

First, put the quadratic into canonical form by dividing by A, if necessary.

Then, take B/2 into a new variable, call it F. Get F^2.

The roots are then -F +- sqrt(F^2 - C)