Question

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

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

Answer 1

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

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 ${\alpha }^2+\alpha +b=0$

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

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

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

$\Rightarrow b^2=-3$ or $b=\sqrt{3}i$