Given a second order Differential Equation, how does one derive the Characteristic equation where one can evaluate and find the constants

Given a ODE of the 2nd order, Ay''+by'+cy = 0, we assume the general solution of the exponential form y=e^(mx).As we will see this leads to an easy simplification due to the properties of the exponential . From this we substitute in and we get Am^(2)(e^mx) +bm(e^mx) + c(e^mx) = 0 here we have a like term of e^mx and thus can be eliminated leaving a quadratic of the form Am^2 + Bm + C = 0 where for a particular ODE we can solve quadratically and will have two values of m for a well-defined solution of the ODE.