Question

If $x^2+ax+b=0$ and $x^2+bx+a=0,a\ne b,$ have a common root $\alpha$, then

• A. $a+b=1$
• B. $\alpha +1=0$
• C. $\alpha =1$
• D. $a+b+1=0$

Answer 1

Answer: (C), (D)

The correct options are
C $\alpha =1$
D $a+b+1=0$

Given,
$x^2+ax+b=0$ and $x^2+bx+a=0$, $a\ne b$ have a common root.
$\Rightarrow {\alpha }^2+a\alpha +b=0$
${\alpha }^2+b\alpha +a=0$
On substracting the above two equations we get,
$\alpha (a-b)+(b-a)=0$
$\Rightarrow \alpha =1$
Substituting $\alpha =1$, we get $a+b+1=0$.
Hence, options 'C' and 'D' are correct.