Question

If the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root and $a\ne b,(a,b\in R)$, then the value of $|a+b|$ is

Answer 1

$x^2+ax+b=0...\left(1\right)$
$x^2+bx+a=0...\left(2\right)$

Let the common root be $\alpha$.
So, $\alpha$ will satisfy above two equations.
${\alpha }^2+a\alpha +b=0...\left(3\right)$
${\alpha }^2+b\alpha +a=0...\left(4\right)$

Subtracting $\left(4\right)$ from $\left(3\right)$, we get
$\alpha (a-b)=a-b$
$\Rightarrow \alpha =1(\because a\ne b)$
Put $\alpha =1$ in eqn$\left(1\right),$ we get
$a+b=-1$
$\Rightarrow |a+b|=1$