G5.2 Given that the roots of $x^{2}+a x+2 b=0$ and $x^{2}+2 b x+a=0$ are both real and $a, b>0$. If the minimum value of $a+b$ is $Q$, find the value of $Q$.
$$
\begin{array}{l}
a^{2}-8 b \geq 0 \text { and } 4 b^{2}-4 a \geq 0 \\
a^{2} \geq 8 b \text { and } b^{2} \geq a \\
\Rightarrow a^{4} \geq 64 b^{2} \geq 64 a \\
\Rightarrow a^{4}-64 a \geq 0 \\
\Rightarrow a\left(a^{3}-64\right) \geq 0 \\
\Rightarrow a^{3} \geq 64 \\
\Rightarrow a \geq 4
\end{array}
$$

Minimum $a=4, b^{2} \geq a$
$$
\Rightarrow b^{2} \geq 4 \Rightarrow \text { minimum } b=2
$$
$Q=$ minimum value of $a+b=4+2=6$