Something similar: If you want to write down a formula for solving a cubic: ax^3+bx^2+cx+d = 0, for give a,b,c,d, you can't avoid using complex numbers. This

Something similar: If you want to write down a formula for solving a cubic: ax^3+bx^2+cx+d = 0, for give a,b,c,d, you can't avoid using complex numbers. This is true even if you only care about cubics which have real solutions. To clarify that last sentence, contrast it with the quadratic formula, which has no mention of the imaginary unit i. It's true that a quadratic with real coefficients might have only complex solutions, but you might just interpret that as having no solutions: once the formula tells you "take the square root of a negative", you know it has no real solutions. Maybe this could be worked into something similar to the high school algebra problem, but with a list of axioms about real numbers.