The value of b for which the equations x^2+b x-1=0, x^2+x+b=0 have...

If $\quad a_1 x^2+b_1 x+c_1=0$ and $a_2 x^2+b_2 x+c_2=0$ have a common real root, then $$\begin{aligned} \Rightarrow \quad\left( a_1 c_2-a_2 c_1\right)^2=\left( b_1 c_2-\right. & \left. b_2 c_1\right) \\ & \left( a_1 b_2-a_2 b_1\right) \end{aligned}$$\left.\therefore $$\begin{array}{l}x^2+b x-1=0 \\ x^2+x+b=0\end{array}$$\right\}$have a common root.$\Rightarrow \quad(1+b)^2=\left(b^2+1\right)(1-b)\Rightarrow b^2+2 b+1=b^2-b^3+1-b\Rightarrow \quad b^3+3 b=0 \Rightarrow b\left(b^2+3\right)=0\Rightarrow \quad b=0, \pm \sqrt{3} i$