Question

The value of $a+b$ if the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root is

Answer 1

Let $\alpha$ be a common root of the equations.
Then $\alpha$ will satisfy the equations.
Thus,
$\alpha^2+a.\alpha+b=0\cdots(i)$
$\alpha^2+b.\alpha+a=0\cdots(ii)$
Subtracting the two equations, we get:
$\alpha(a-b)+(b-a)=0$
$\Rightarrow \alpha(a-b)-(a-b)=0$
$\Rightarrow \alpha(a-b)=(a-b)$
Since \(a\ne b\)
$\therefore \alpha=1$
Hence, placing the value of $\alpha$ in the equation (i), we get
$1^2+a.1+b=0\Rightarrow a+b=-1$
Hence, value of $a+b=-1$