Question

A value of b for which the equations
$x^2+bx-1=0x^2+x+b=0$
have one root in common is -

• A. $-\sqrt{2}$
• B. $-i\sqrt{3}$
• C. $i\sqrt{5}$
• D. $\sqrt{2}$

Answer 1

The correct option is B $-i\sqrt{3}$

Let $\alpha$ be the common root of given equations, then
${\alpha }^2+b\alpha -1=0.......\left(1\right)$
and ${\alpha }^2+\alpha +b=0.......\left(2\right)$
Subtracting (2) from (1), we get
$(b-1)\alpha -(b+1)=0$
or $\alpha =\frac{b+1}{b-1}$
Substituting this value of $\alpha$ in equation (1) , we get
$(\frac{b+1}{b-1}{)}^2$ +$b\left(\frac{b+1}{b-1}\right)-1=0$
or $b^3+3b=0$
$\Rightarrow b=0,i\sqrt{3},-i\sqrt{3}$