Question

A value of b for which the eqn $x^2+bx-1=0,x^2+x+b=0$ 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 $i\sqrt{3}$

Given:$x^2+bx-1=0$ and $x^2+x+b=0$ have one common root.

Let the common root be $\alpha$

$\Rightarrow {\alpha }^2+b\alpha -1=0$ and ${\alpha }^2+\alpha +b=0$

$\Rightarrow \frac{{\alpha }^2}{-1-b^2}=\frac{-\alpha }{-1-b}=\frac{1}{b-1}$

$\Rightarrow \alpha =\frac{b+1}{b-1},\alpha =-\frac{b^2+1}{b+1}$ and ${\alpha }^2=\frac{b^2+1}{1-b}$

$\Rightarrow {(1+b)}^2=(b^2+1)(1-b)$

$\Rightarrow b^2+1+2b=b^2-b^3+1-b$

$\Rightarrow b^3+3b=0$

$\Rightarrow b(b^2+3)=0$

$\Rightarrow b\ne 0,b^2=-3$

$\therefore b=\sqrt{-3}=\sqrt{3}i$