Question

For $a\ne b,$ if the equation $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root, then the value of $a+b$ equals to:

• A. $-1$
• B. 0
• C. 1
• D. 2
• E. $-2$

Answer 1

Answer: (A)

Let $\alpha$ be the common root for both the equations $x^2+ax+b=0$ and $x^2+bx+a=0,$ then
${\alpha }^2+a\alpha +b=0$ ...(i) and
${\alpha }^2+b\alpha +a=0$ ...(ii)
$\Rightarrow$ $\frac{{\alpha }^2}{\left|\begin{array}{cc}a& b\\ b& a\end{array}\right|}=\frac{\alpha }{\left|\begin{array}{cc}b& 1\\ a& 1\end{array}\right|}=\frac{1}{\left|\begin{array}{cc}1& a\\ 1& b\end{array}\right|}$
$\Rightarrow$ $\frac{{\alpha }^2}{(a^2-b^2)}=\frac{\alpha }{b-a}=\frac{1}{b-a}$
$\therefore$ ${\alpha }^2=-(a+b)$ and $\alpha =1$
Hence, $a+b=-1$