Leaving aside the fact that this isn't new -- it's how I was taught to solve quadratic equations in the 80s -- if you look closely he's still completing the square.
When I saw the title I was expecting some kind of geometric approach which was worrying to me because I think people are even worse at understanding geometric proofs of anything like this than they are at understanding algebraic proofs. Indeed, completing the square is a geometric technique originally and that is, I think, why it has that name (which makes no sense to the high school students who are taught it).
I’m also struggling to see the innovation in the article, or rather why it is any better, but I think a lot of my problem is that I’m basically fine at all the abstract manipulation of symbols required for school algebra and was when learning about completing the square yet I think it is one of the fundamentals that people really struggle with. When you look at the derivation of the quadratic formula, you need relatively complex expressions and I think only the strongest high school students at algebra (at least from my school) will be able to cope with the derivation. But I think this new method doesn’t help with what I think are the most common confusions: not understanding what a variable is (or variable notation) and not understanding what a function is (or function notation; though I think in this case it maybe doesn’t help that the function is implicit). I feel like I’m just a bad person to judge this method (but I think most people capable of talking about it would be poor judges). I’d be interested in the results of an experiment that tried different methods of teaching but the problem with such experiments is that the teacher in the experiment is likely to understand the method much better than a random teacher outside of the experiment.