Quadratic and Cubic equations

About two weeks ago my son asked me a fun question about cube roots.  We’d spent a lot of time studying quadratic equations over the last couple of months, but for some reason he was looking at cube roots rather than square roots this morning (I don’t remember why).  The problem he was working on required him to find the cube root of 343.  He asked me why there was only one solution here but square roots had two solutions.

I’d been looking for something to talk about over Christmas break anyway, and this random little question definitely was ready made for a week of talks.  We spent a total of seven days discussing quadratics and cubics and had a lot of fun.

(1) The first day we spent some time playing around on Mathematica looking at the difference between graphs of quadratic equations and graphs of cubic equations  (with real coefficients).  After doing that, we talked about what we saw:

(2) The next day we went back to look at a little algebra.  The algebra book we are studying (Art of Problem Solving’s “Algebra”) had sections on differences of squares and  differences of cubes.  We’d spent so much time away from cubes, though, that I didn’t expect that my son remembered much about differences of cubes.  We revisited that topic in this video and show how we can use the quadratic formula to help us find all of the roots of an equation that is a difference of cubes:

(3) In the last video we found the three roots of the equation x^3 = 1.  In this video I tried to show a little bit of geometry behind these numbers.  I wanted to do talk through some of the geometry before doing a bit more arithmetic with the imaginary numbers.   We also ended up spending a lot of time in this video talking about the equation x^4 = 1.

(4) Here we dig into the geometry of the solutions of x^3 = 1 a little more.  In particular, we check that the three solutions all lie on a circle of radius 1 in the imaginary plane.

(5) Next up, we finally get around to checking whether or not the three solutions we found to x^3 = 1 are actually solutions.  So, this video is a little bit of arithmetic on complex numbers.

(6) Having now gone through quite a bit of algebra, geometry, and arithmetic, I wanted to spend a little time on history.  The point was to give a little context to the question my son asked and explain that polynomials with degree “n” have n solutions.  The fact that we find two solutions for quadratics and three solutions for cubics was just a special case of a more general fact.

(7) I didn’t feel like we should end on the fundamental theorem of algebra, so we did one more little talk.  In this one we talk through the history of finding solutions to polynomials.  We start with the quadratic equation and end with Galois.

All in all, this was a fun way to spend time on Christmas break.  Happy that my son asked me about cube roots last week!