Question

If the equations $a{x}^2+bx+c=0$ and $c{x}^2+bx+a=0$ have one root in common, prove that $a+b+c=0ora-b+c=0$.

Answer 1

If equation $a{x}^2+bx+c=0$ & $c{x}^2+bx+a=0$ have a common root prove $a+b+c=0$ or $a+b+c=0$

$\to$ Let assume $\alpha$ is common root

put in equation $\left(i\right)$ & $\left(ii\right)$

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

$\therefore$ Compare both the equation

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

$\therefore a{\alpha }^2-c{\alpha }^2=a-c$ Substitute the value of $\alpha$ in equation

$\therefore {\alpha }^2(a-c)=a-c$

$\therefore {\alpha }^2=1$ $\therefore$ we get, $a+b+c=0(\alpha +1)$ & $a-b+c=0$(for $\alpha =-2$)

$\therefore \alpha =+1,-1$