# Using Po-Shen Loh’s quadratic formula idea to calculate the Cosine of 72 degrees

We did a project using the ideas of sums and products of roots at the time, and I wanted to revisit the idea tonight now that my son is studying complex numbers. An idea I thought would be fun was to explore was how to use the sum and product of roots ideas for quadratic functions to calculate the cosine of 72 degrees.

I started with a quick review of the main ideas we’d be using in this project as it has been several months since we went through Po-Shen Loh’s idea:

Now we dove into the problem of finding the roots of the equation $x^5 - 1 = 0$.

Now we moved to the main idea in the project – how can we factor the polynomial we found in the last video – $x^4 + x^3 + x^2 + x + 1$ – into two quadratic polynomials?

The work here is a little tricky, but my son got through it really well. The ideas here are definitely accessible to students who have learned a little bit about polyomials and sums and products of roots.

Finally, we solved for the roots of the quadratic equation $x^2 + x - 1 = 0$ (I accidentally wrote this equation wrong, so we get off to a bad start. Luckily we caught the error after about a minute.)

Solving this equation gives us the value of cos(72)!

It was really fun to see that the combination of introductory ideas from complex numbers, polynomials, and sums / products of roots of quadratics could help us calculate the value of cos(72). I’m excited to play around with Po-Shen Loh’s idea a bit more and see where else we can find some fun applications!