The amplitude of vibration of a particle is given by  $a_m=\frac{a_o}{a{\omega }^2-b\omega +c}$, where $a_o$, a, b & c are positive. The condition for a single resonant frequency.

$$\begin{gathered} for \sin gle resonant frequency \\ a{w}^2-bw+c=0 \\ w = \frac{+b\pm \sqrt{b^2-4ac}}{2a} \\ for \sin gle value of resonant frequency the condition is discriminant =0, \\ b^2-4ac=0 \\ \Rightarrow b^2=4ac \end{gathered}$$