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Very nice article! Was looking for some material to inspire high school student to solve quadratics, and this is super helpful!
One comment though  I don't quite agree with the portayal of imaginary numbers. There is no "cheating" involved in defining a number system where i = sqrt(1). The only reason x^2 > 0 for real x is because we have _defined_ multiplication that way. We ourselves make these rules about number systems, and we're free to make new ones; hence, the imaginary numbers are not more strange or "illegal" than anything else in mathematics. At some point in time, people were also uncomfortable with the idea of subtracting a larger number from a smaller one, because it was felt that negative numbers "don't exist". And the same can be said for irrational numbers.
Personally, I think the terminology "real" and "imaginary" numbers is unfortunate. The truth is of course that *all* numbers are imaginary; they exist only in our imagination!