If the quadratic equations $x^2+ax+b=0$  and $x^2+bx+a=0$  have a common root then

$\to$ let $\text{'}\alpha \text{'}$ be common root

${\alpha }^2+a\alpha +b={\alpha }^2+b\alpha +a=0$

$a\alpha -b\alpha =a-b$

$\alpha \left(a-b\right)=x-b$

$\alpha =1$

(1) $\Rightarrow 1^2+a+b=0\Rightarrow a+b=-1$