The amplitude of vibration of a particle is given by am=aoaω2−bω+c, where ao, a, b & c are positive. The condition for a single resonant frequency.
b2 = 4ac
b2 > 4ac
b2 = 5ac
b2 < 4ac
for single resonant frequency aw2-bw+c=0 w = +b±b2-4ac2a for single value of resonant frequency the condition is discriminant =0, b2-4ac=0 ⇒b2=4ac