The amplitude of vibration of a particle is given by am=(a0)/(aω2−bω+c); where a0,a,b and c are positive. The condition for a single resonant frequency is

For resonance amplitude must be maximum which is possible only when the denominator of expression is zero  i.e.  aω2−bω+c=0⇒ω=-b±b2−4ac2aFor a single resonant frequency, b2 = 4ac.