The first is a simple one, do we use the infinite sum of the power series (xⁿ) or the exponential series (_exⁿ).

They both do the job, but each has computational problems at very big (n x 10^m) and very small (n x 10^-m) numbers.

Not all solutions in this involve big or small numbers; sometime, and that's when the result goes wild.

Second is the solution to the general Cubic Equation ax³ + bx² + cx¹ + dx⁰ = 0.

The roots of the equation involve some imaginary or complex numbers and result in a solution that needs a choice with only real numbers.

Seems the real solution jumps between the roots of x₁, x₂, and x₃.

see http://en.wikipedia.org/wiki/Cubic_equation

and http://mathworld.wolfram.com/CubicFormula.html

Some our solved equations have the roots as the soltiuon, but it does not remain fix to any one root.

We've been looking for a way to determine much more accurately the outcome from the roots, but it's a very complex solution to work with.