It’s certainly no harder than any other way of describing complex solutions to quadratic equations. (But you probably want to start with quadratic equations that have scalar solutions.)
It’s quite straight-forward to show graphically what it means to square an origin-preserving similarity transformation of the plane: just apply it twice. So if your original complex number transformed an arbitrary chosen vector by scaling it some amount and rotating it some amount, just scale and rotate the result again. This is no mostly the same idea as repeated multiplication by scalars, as taught in primary school, but with planar rotations thrown in.
Adding complex numbers can be shown graphically by demonstrating what effect the sum has on a vector (since the geometric product distributes over addition), or where convenient what effect it has on a square grid.
We can easily graphically apply some quadratic equation to arbitrary complex numbers, and see what it does to them (again, where we draw the complex numbers using points o + zv for some arbitrarily chosen origin o and “unit vector” v).
Showing that when we take a quarter-turn in our plane I, and apply it twice, we get II = –1 is a relatively trivial observation. It relies on the observation that a half-turn rotation is equivalent to a reflection across the axis of rotation. So that gives us I as one of the roots to xx + 1 = 0.
The root of a quadratic equation is just the complex number which gets mapped to 0. So try applying your quadratic equation to different complex numbers and see where they go. There will always be two complex numbers mapping to zero (except in the case of a double root). Once we have some intuition for the problem we can try to develop some concrete algorithms or strategies for finding those roots efficiently. We can also try to make other kinds of plots to help us, e.g. using a tool like https://observablehq.com/d/3a81942cd20f81e4
I haven’t tried teaching quadratic equations to middle schoolers, but since the graphical version mostly entails drawing pictures, I would guess it to be rather easier than a purely algebraic version. Either way there are quite a lot of tricky concepts to explore and unpack, so the students are going to need months of sustained effort.
This post shows clearly that geometric intuition (other, perhaps, than drawing a parabola crossing - or not crossing - the X-axis) is not helpful at all understanding and solving a quadratic equation, while elementary algebra indeed can and does; the imaginary unit, then, appears naturally as the square root of the negative unit (which leads to the need to extend the reals); if anything, the approach that you advocate would make the kid's head spin (no pun intended) with the result being the total loss of the ability to think about things in a simple and natural way.
I said essentially «there are many concepts involved which will take months to unpack, irrespective of which approach is taken, but in good faith here are some sketchy descriptions of what pictures you might draw if you wanted to start with a graphical interpretation ...»
To which you responded essentially «your sketch summary consisting of a few picture-free paragraphs aimed at an expert audience doesn’t completely describe all of a complicated set of ideas to a naïve middle school student anywhere near as well as 2 semesters of introductory algebra class does at covering a more limited subset of the same ideas, therefore it must be useless.» This after you earlier proposed an approach to complex numbers that optimistically will first be accessible for typical 3rd year undergraduate math majors.
This conversation is absurd. But all the best to you.
It’s quite straight-forward to show graphically what it means to square an origin-preserving similarity transformation of the plane: just apply it twice. So if your original complex number transformed an arbitrary chosen vector by scaling it some amount and rotating it some amount, just scale and rotate the result again. This is no mostly the same idea as repeated multiplication by scalars, as taught in primary school, but with planar rotations thrown in.
Adding complex numbers can be shown graphically by demonstrating what effect the sum has on a vector (since the geometric product distributes over addition), or where convenient what effect it has on a square grid.
We can easily graphically apply some quadratic equation to arbitrary complex numbers, and see what it does to them (again, where we draw the complex numbers using points o + zv for some arbitrarily chosen origin o and “unit vector” v).
Showing that when we take a quarter-turn in our plane I, and apply it twice, we get II = –1 is a relatively trivial observation. It relies on the observation that a half-turn rotation is equivalent to a reflection across the axis of rotation. So that gives us I as one of the roots to xx + 1 = 0.
The root of a quadratic equation is just the complex number which gets mapped to 0. So try applying your quadratic equation to different complex numbers and see where they go. There will always be two complex numbers mapping to zero (except in the case of a double root). Once we have some intuition for the problem we can try to develop some concrete algorithms or strategies for finding those roots efficiently. We can also try to make other kinds of plots to help us, e.g. using a tool like https://observablehq.com/d/3a81942cd20f81e4
I haven’t tried teaching quadratic equations to middle schoolers, but since the graphical version mostly entails drawing pictures, I would guess it to be rather easier than a purely algebraic version. Either way there are quite a lot of tricky concepts to explore and unpack, so the students are going to need months of sustained effort.