We review the standard formulas for solving quadratic and cubic equations, the latter going back to work in the 1500's by del Ferro, Tartaglia and Cardano, and pointing out that in reality these formulas only generate approximate solutions, at least in general. We also connect both formulas with Newton's method to solve an example quadratic and an example cubic, at least approximately.This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.