The ratio of the roots of  $a{x}^2+bx+b=0$ is  $p:q$, then

$\alpha +\beta =-\frac{b}{a};\alpha \beta =\frac{b}{a}\text{. As }\alpha :\beta =p:q\text{, take }\alpha =p\lambda \text{and }\beta =q\lambda \text{.}$

$\Rightarrow (p+q)\lambda =-\frac{b}{a}\text{and }pq{\lambda }^2=\frac{b}{a}$

$\Rightarrow \lambda =\frac{\sqrt{b}}{\sqrt{apq}}\text{and}(p+q)\frac{\sqrt{b}}{\sqrt{apq}}=-\frac{b}{a}$

$\Rightarrow \sqrt{\frac{bp}{aq}}+\sqrt{\frac{bq}{ap}}+\frac{b}{a}=0\Rightarrow \sqrt{\frac{p}{q}}+\sqrt{\frac{q}{p}}+\sqrt{\frac{b}{a}}=0$