Find the minimum of $a+b$

For both the equations to have real roots, we have:

$\begin{cases} x^2+ax+2b = 0 & \implies a^2 \ge 8b & ...(1) \\ x^2+2bx+a = 0 & \implies 4b^2 \ge 4a \implies b^2 \ge a & ...(2) \end{cases}$

$\begin{aligned} (1)^2: \quad a^4 & \ge 64\color{#3D99F6}{b^2 \quad \quad \small \text{From }(2): b^2 \ge a} \\ & \ge 64\color{#3D99F6}{a} \\ a^3 & \ge 64 \\ \implies a & \ge 4 \\ (2): \quad \ \ b^2 & \ge 4 \\ \implies b & \ge 2 \\ \implies a + b & \ge \boxed{6} \end{aligned}$