Hidden Polynomial Equations and Substitution (C)

Just as there are hidden quadratic equations, there are also hidden polynomial equations. These are best solved by substitution. Often the substitution will contribute a side condition. For example, to find all real numbers x so that

e^3x - 4e^2x + 5 = 0

we introduce the substitution u = e^x which introduces the side condition u > 0 to go along with the cubic equation

u³ -4u² + 5 = 0.

Since u³-4u² + 5 = (u+1)(u² - 5u+5), the cubic equation in u can be solved completely. However, there is no x corresonding to u = -1 since e^x = -1 has no solution for x a real number.