Series RLC circuit problem
what it means by "the ckt is in quiescent state" ?


It means that there was no energy in either the inductor or capacitor prior to the voltage being applied.


  why is he getting sqrt(8) in the current equation ?


With an inductor of 0.5 henries and a capacitor of 0.25 farads, the natural resonant frequency is \$\sqrt8\$ radians per second.

This is from the equation \$\omega_n = \dfrac{1}{\sqrt{LC}} \$

In other words there will be a damped oscillation in the current starting at a peak and decaying with a natural frequency of \$\sqrt8\$ radians per second.