The quadratic equation of the form $x^2 + ax + b = 0$ 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 $e_1 \textrm{ and } e_2$ be two (not necessarily different) solutions to $x^2 + ax + b = 0$. By the fundamental theorem of algebra $(x - e_1)(x - e_2) = x^2 + ax + b$. Multiplying out $x^2 -(e_1 + e_2) x + e_1e_2=x^2 + ax + b$. So $e_1 + e_2 = -a \textrm{ and } e_1e_2 = b.$ This is a non-linear system of equations in two variables.
Note that ${(e_1 - e_2)}^2 = e_1^2 + e_2^2 -2e_1e_2$ which is equal to ${(e_1 + e_2)}^2 - 4e_1e_2 .$ Thus $e_1 - e_2 = {(a^2 - 4b)}^{1/2}$ Which easily yields the solution $e_1 = -a/2 +- {(a^2 - 4b)}^{1/2}.$
It’s a fun exercise. The trick is writing the square of $e_1 - e_2$ and also assuming you already have two solutions.