If $a{x}^2+bx+c=0,\text{\hspace{0.17em}\hspace{0.17em}and }b{x}^2+cx+a=0$ have a common root and $a\ne 0,b\ne 0,c\ne 0$, then $\frac{a^3+b^3+c^3}{abc}$ is 

$a{x}^2+bx+c=0,\text{\hspace{0.17em}\hspace{0.17em}and }b{x}^2+cx+a=0$  have a common root

${(bc-a^2)}^2=(ab-c^2)(ac-b^2)$

$b^2{c}^2+a^4-2a^{2}bc=a^{2}bc-a{b}^3-a{b}^3+b^2{c}^2$

$a^4+a{b}^3+a{c}^3=3a^{2}bc$

$a(a^3+b^3+c^3)=3abc\times a$

$a^3+b^3+c^3=3abc$