If every pair from among the equations $x^2+ax+bc=0$ , $x^2+bx+ca=0$ and $x^2+cx+ab=0$ has a common root, then

$x^2+bx+ca=0\to \left(2\right)$

$x^2+cx+ab=0\to \left(3\right)$

$\alpha$  is a common root

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

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

$\left(a-b\right)\alpha +c\left(b-a\right)=0$

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

$\alpha =c$

For 1, 2 common root c

For 2, 3 common root a

For 3, 1 common roots b

Sum of 3 common roots $=b+c+a$

Product of common roots is $abc$