The discriminant — From a quadratic equation to dynamic nonlinear systems
Most mathematics students don't understand the potential of the discriminant. Usually discussed while learning the Quadratic Formula to solve quadratic equations, students learn its use for the classification of type and number of solutions for each equation being solved. Most students memorize the formula, identify the correct values for each variable, and successfully solve to find solutions for the equation, but few of them ever understand or appreciate the power of that piece of the formula called the discriminant. There are many additional applications for the discriminant, and I will explore these. In particular, I discuss the role of the discriminant in: · The Quadratic Formula · Solution of quadratic equations with parameters · Solving linear differential equations with constant coefficients · Investigating types of solutions for linear differential equations with variable coefficients · Applications in Number Theory to Diophantine equations · Classifying conic sections · Dynamic nonlinear systems.