When Formulas Fail

Much of high school mathematics makes one believe there is a formula for everything, yet those pursuing higher level mathematics quickly find out that this is not the case. They also find that the world of mathematics is far larger than they ever imagined. This task will give you a glimpse into that world.

  1. Complete the square to find the roots of the following:Screen Shot 2020-04-22 at 5.59.39 PMAssuming a, b, and c are constants, complete the square to find the roots of the following:Screen Shot 2020-04-22 at 5.59.59 PMWhat do you notice about your answers to question 2? What famous formula does this look like?
  2. You’ve probably figured out that you derived the quadratic formula in problem 2. But is there a cubic formula? First, find the roots of the following cubic, knowing that all three are integers:Screen Shot 2020-04-22 at 6.03.37 PM
  3. It turns out there is a cubic formula, but deriving it is a nightmare. It can be written in the two forms below, but take some time to verify that they are, in fact, the same. Then, use one of the formats to find the routes of the cubic in question 5 to verify your answers:
  4. It turns out there is an even messier quartic formula for degree four polynomials, which you can read about more here if you choose. However, you will not be asked to use it or derive it. Even crazier, it turns out there is no formula for polynomials of degree five or higher based on the Abel-Ruffini Theorem. In order to understand this though, you will have to take an abstract algebra course, which is usually a senior level university course. All of this is to set you up for the fact that, although there may not be exact formulas, mathematicians have devised numerical methods for finding routes without a formula, that are often more practical for scientists and engineers to use for applications. Work through the methods found in this article and then test them on polynomials of your own creation.