A new way to make quadratic equations easy (2019)

To teach quadratics you can use Cavalieri's parabolograph:

https://en.etudes.ru/models/conic-sections-cavalieri-parabol...

This may make all of the variables more "real" - but note that the equation graphed is really A(X - B)^2 = Y - C, where (B, C) is the vertex and A is the "setting" (clear if you examine the device. This is relatively similar to the complete-the-square formalism but with a straightforward geometric and "hands-on" interpretation.

That is just one equation. That is always easy

I need to solve quadratic equations. It is much harder. Like

a1 x y + b1 x + c1 y + d1 = 0

a2 y z + b2 y + c2 z + d2 = 0

a3 z x + b3 z + c3 x + d3 = 0

Given a,b,c,d, solve it for x,y,z.

And in the general case where there are more than 3 equation or more than 3 x,y,z variables.

We can make a undirected graph with the variables as nodes and an edge if there is an equation containing those two variables. Like (x,y), (y,z), (x,z) here. If there is a cycle in that graph, we can combine the equations to obtain a single quadratic equation, which has two solutions for x.

I can search for cycles, and sometimes two cycles restrict each other, such that only one solution for x remains. Sometimes two solutions remain. I do not understand when that happens, and if there is a better way than searching cycles.

(also I can have an additional list of restrictions between the parameters. Like a1 = -c2. This sometimes leads to a cycle canceling itself out, revealing no information about x at all. )

The author should really remove the words 'simple', 'easy', and 'intuitive' from this article. They clearly know nothing of why most people detest math. If you're already familiar with the quadratic equation and you care about math and even enjoy it you might find this new equation interesting, useful, or intuitive, but I guarantee it is still a garbled mass of un-intuitive numbers and symbols to "[m]any former algebra students [who] have painful memories of struggling to memorize the quadratic formula."

No he should not. The words invokes the right connotations in a pedagogically minded reader. This can be the basis for better teaching. It depends of course on implementation.

I briefly looked at the author's paper. I find the account of the traditional method there questionable. He suggests it starts by going from:

  x^2 + bx + c

to

  x^2 + bx + c + b^2/4 - b^2/4

This step is freaking weird, I've never seen it before, and I don't think this way. The process I follow is more like I observe that for general k:

  (x + k)^2 = x^2 + 2kx + k^2

so I observe that for k = b/2 I get

  (x + b/2)^2 = x^2 + bx + b^2/4

and then I figure out what I need to add to both sides of that equation to make the RHS equal the original quadratic, which gives me:

  (x + b/2)^2 - b^2/4 + c = x^2 + bx + c

and I then find the roots of the RHS by instead finding them for the LHS. We say that the LHS is the result of "completing the square" on the RHS.

Question: Does anyone seriously learn to solve quadratics by rewriting x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4. Because that seems like a bad way of learning it.