Quadratic Equation

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.

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