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

Let $\alpha$ be the common root.

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

$$\begin{gathered} \Rightarrow \left|\begin{array}{cc}1& b\\ 1& 1\end{array}\right| x \left|\begin{array}{cc}b& -1\\ 1& b\end{array}\right|={\left|\begin{array}{cc}-1& 1\\ b& 1\end{array}\right|}^2 \\ \Rightarrow (1-b)(b^2+1)={(-1-b)}^2 \\ \Rightarrow b^3+3b=0 \end{gathered}$$

$\therefore b=0, i\sqrt{3}, -i\sqrt{3},$where $i=\sqrt{-1}$