I don’t see the “innovativeness” of this method. Solving by factorization is commonly taught in Algebra 1 (at least in US schools) as an alternative to the quadratic equation (see Khan academy for a refresher).
The problem with solving by factoring is that it doesn’t really work well when the x^2 coefficient is not 1 or when the values are fractions or irrational numbers because factorizing this way requires to find a pair of numbers that sums and multiplies to a set of two given numbers. In this case, you will need to use the quadratic equation.
If you are interested in a cool way to make factoring a quadratic equation simpler, CPM covers a good visual way to organize that work.
The paper shows a simple way to get the quadratic formula from remembering few simple rules you should remember anyway. I dunno if there’s innovation there but it’s nice and removes the mechanical rote-memorization that rots school mathematics. That CPM link btw. is a mess.
Presentation and expectations about this aside, it teaches a really important technique that took me way to long to pick up in mathematics: guess at what the solution looks like and work backwards.
It’s based on knowing the fundamental theorem of algebra to come up with the (x - A) (x - B) formulation, but you can suspect that will be the shape of solutions to problems like this long before you get around to proving the fundamental theorem.
(x - A) (x - B)
You don’t even need to assume the FTA – you go through the derivation with the assumption that it’s only valid if the quadratic does have two roots, but once you have the expression for the roots, you can substitute the expression for any quadratic to show that they’re indeed both roots.
Alternatively, you can show that any (real or complex) quadratic has a complex root, which is much simpler than the full FTA. You have to be careful to avoid a circular argument, though, since the quadratic formula already implies this!
I think this video highlights what makes this method easier: https://youtu.be/ZBalWWHYFQc
Is this some kind of joke or parody? ‘This new method that is so much more innovative than the quadratic formula’ is just yet another bog-standard derivation of the quadratic formula???