Question

Find $a$ if equations $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have a common root

Answer 1

Let the common root be $\alpha$

$\Rightarrow {\alpha }^4+a{\alpha }^2+1=0,{\alpha }^3+a\alpha +1=0$

Subtracting both eqns, ${\alpha }^4-{\alpha }^3+a{\alpha }^2-a\alpha =0$

$\Rightarrow \alpha ({\alpha }^3-{\alpha }^2+a\alpha -a)=0$ (but $\alpha =0$ does't give a root)

$\Rightarrow {\alpha }^3-{\alpha }^2+a\alpha -a=0$

$\Rightarrow {\alpha }^2(\alpha -1)+a(\alpha -1)=0$

$\Rightarrow (a+{\alpha }^2)(\alpha -1)=0$

$\Rightarrow \alpha =1$ or $a=-{\alpha }^2$

Clearly $\alpha =1$ gives $a=-2$ in both equations,

hence for $a=-2$ both equations have a common root