$\triangle _{1}=a_{1}^2-4b_{1},\triangle _{2}=a_{2}^2-4b_{2}$Do đó $\triangle _{1}+\triangle _{2}=a_{1}^2+a_{2}^2-4(b_{1}+b_{2})\geq a_{1}^2+a_{2}^2-2a_{1}a_{2}\geq 0$
$\Rightarrow \left[ {\begin{matrix} \triangle _{1}\geq 0\\ \triangle _{2}\geq 0\end{matrix}} \right.\Rightarrow đpcm$