The quadratic equation of the form is one of the simplest nonlinear problems in one variable. The method I use for solving it can also be applied to the cubic and quartic equations but of course not the quintic.
Let be two (not necessarily different) solutions to . By the fundamental theorem of algebra . Multiplying out . So This is a non-linear system of equations in two variables.
Note that which is equal to Thus Which easily yields the solution
It’s a fun exercise. The trick is writing the square of and also assuming you already have two solutions.