Question
A value of $$b$$ for which the equations
$$\eqalign{
& {x^2} + bx - 1 = 0 \cr
& \,{x^2} + x + b = 0 \cr} $$
have one root in common is
A.
$$ - \sqrt 2 $$
B.
$$ - i\sqrt 3 $$
C.
$$ i\sqrt 5 $$
D.
$$ \sqrt 2 $$
Answer :
$$ - i\sqrt 3 $$
Solution :
Let $$\alpha $$ be the common root of given equations, then
$$\eqalign{
& {\alpha ^2} + b \alpha - 1 = 0\,\,\,\,\,\,\,\,\,\,\,.....\left( 1 \right) \cr
& {\text{and }}\,{\alpha ^2} + \alpha + b = 0\,\,\,\,\,\,\,\,\,\,\,\,.....\left( 2 \right) \cr} $$
Subtracting (2) from (1), we get
$$\eqalign{
& \left( {b - 1} \right)\alpha - \left( {b + 1} \right) = 0 \cr
& {\text{or }}\alpha = \frac{{b + 1}}{{b - 1}} \cr} $$
Substituting this value of $$\alpha $$ in equation (1), we get
$$\eqalign{
& {\left( {\frac{{b + 1}}{{b - 1}}} \right)^2} + b\left( {\frac{{b + 1}}{{b - 1}}} \right) - 1 = 0\,\,\,{\text{or}}\,\,{b^3} + 3b = 0 \cr
& \Rightarrow \,\,b = 0,i\sqrt 3 , - i\sqrt 3 \cr} $$