The amplitude of vibration of a particle is given  by  am=a0aω2−bω+c where a0,a,b and c are positive. The condition for a single resonant frequency is

For resonance, Amplitude is maximum (infinity) as denominator function is quadratic equation   ax2+bx+c=0 aω2-bω+c=0Roots of  ω , ω = +b±b2−4ac2a if  ω  = 0 ⇒ b2−4ac=0   b2=4ac