Thank you for reporting, we will resolve it shortly
Q.
A value of $b$ for which the equations $x^2+bx-1=0,x^2+x+b=0$ have one root in common is
IIT JEEIIT JEE 2011Complex Numbers and Quadratic Equations
Solution:
If $a_{1} x^{2}+b_{1} x+c_{1}=0$
and $a_{2} x^{2}+b_{2} x+c_{2}=0$
have a common real root, then
$\Rightarrow\left(a_{1} c_{2}-a_{2} c_{1}\right)^{2}=\left(b_{1} c_{2}-b_{2} c_{1}\right)\left(a_{1} b_{2}-a_{2} b_{1}\right)$ have a common root
$\Rightarrow\left(1+b^{2}\right)=\left(b^{2}+1\right)(1-b)$
$\Rightarrow b^{2}+2 b+1=b^{2}-b^{3}+1-b$
$\Rightarrow b^{3}+3 b=0$
$\therefore b\left(b^{2}+3\right)=0 \Rightarrow b=0, \pm \sqrt{3} i$
have a common root