Series RLC Max Voltage calculation Problem
I'd solve the transfer function H(s) with the capacitor across the output (as per a 2nd order low pass filter) then move on to solve for L at the output. With C at the output you have: -



So, if you do a little algebra, \$H(s) = \dfrac{1}{s^2LC+sCR+1}\$

Then put it into the standard format to solve for s: -

\$H(s) = \dfrac{\frac{1}{LC}}{s^2+s\frac{R}{L}+\frac{1}{LC}}\$

Maximum is when s = 0 so using the quadratic solution for s....

\$s = \frac{-R}{2L}+/-\sqrt{\frac{R^2}{4L^2}-\frac{1}{LC}}\$

Now if you know your peaking filters the terms after the +/- are complex conjugate poles aligned with frequencies of: -

+/-  \$j\sqrt{\frac{1}{LC}-\frac{R^2}{4L^2}}\$.

It looks like your answer is missing an \$L^2\$ - you just have "L" without it being squared. You also have 2 in front of the L instead of 4: -



Thanks to jonk for spotting my error