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General

The standard form of the characteristic equation   is

$$s^2 + 2 \zeta \omega_0 s + \omega_0^2 = 0$$ (38)

where

$\omega_0$ is the natural frequency.  

$\zeta$ is the damping ratio. Critical damping corresponds to $\zeta =1$.   

$s_{1,2} = (-\zeta \pm \sqrt{\zeta^2-1})\omega_0$ are the two roots.

In our example, $\omega_0 = 2$ rad/sec, and $\zeta=R/4$.