Solving RLC Circuit Problem: Underdamped CKT Roots

[Isma]

i have a question where
s1,2=(-4 (+-) j3) 10^3
are roots of a second order D.E of a series RLC ckt
where s1,2=(-z (+-) (z^2 -1)
(z is zeta)
now from
s1,2=(-4 (+-) j3) 10^3 ,its showing roots of underdamped ckt(complex nd conjugate)
but when i solve eq. z gives 2 values(but z must be less than 1 nd gr8er than 0 acc. 2 condition of underdamped ckt)
help please

 

[SGT]

i have a question where
s1,2=(-4 (+-) j3) 10^3
are roots of a second order D.E of a series RLC ckt
where s1,2=(-z (+-) (z^2 -1)
(z is zeta)
now from
s1,2=(-4 (+-) j3) 10^3 ,its showing roots of underdamped ckt(complex nd conjugate)
but when i solve eq. z gives 2 values(but z must be less than 1 nd gr8er than 0 acc. 2 condition of underdamped ckt)
help please

The roots of a second order system are:
$$s_{1,2} = -\zeta \omega_n \pm j\omega_n \sqrt{1 - \zeta^2}$$
where $$\omega_n$$ is the undamped natural frequency and $$\zeta$$ is the damping coefficient.