Question

Show that the equation $a{x}^2+bx+c=0$ and $x^2+x+1=0$ cannot have a common root unless $a=b=c$

Answer 1

Given that,

$x^2+x+1=0$ has non real roots.

Hence, $a{x}^2+bx+c=0$ and above equation has common roots if

$\begin{array}{c}\frac{a}{1}=\frac{b}{1}=\frac{c}{1}\\ \Rightarrow a=b=c\end{array}$