By using the quadrature method, we study the exact number of positive solutions of the following quasilinear boundary value problem: $$ \align -\big(\varphi_p(u')\big)'&=\lambda f(u)\text{ in }(0,1),\\ u&>0\text{ in }(0,1),\\ u(0)=u(1)&=0, \endalign $$ where $\varphi_p(y)=|y|^{p-2}y$, $y\in\Bbb R$, $p>1$, $\lambda>0$ and $f:\Bbb R_+ \to \Bbb R_+$ is of class $C^2$ and $p$-convex function.