Question

If the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root, then the value of $a+b$ is:

• A. $+1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

The correct option is C $-1$

Given: equations $x^2+ax+b=0$ and $x^2+bx+a=0$ having a common root

To find the value of $a+b$

Sol: Let the common root be $\alpha$

now, ${\alpha }^2+a\alpha +b=\mathrm{0....}\left(i\right)$

and ${\alpha }^2+b\alpha +a=\mathrm{0....}\left(ii\right)$

By (i)-(ii), we get

$a\alpha -b\alpha +b-a=0⟹\alpha (a-b)=(a-b)⟹\alpha =1$

Put the value of $\alpha$ in any of the given two equations we get

$1^2+a\left(1\right)+b=0⟹a+b=-1$