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.
$\Rightarrow \quad(1+b)^2=\left(b^2+1\right)(1-b)$
$\Rightarrow b^2+2 b+1=b^2-b^3+1-b$
$\Rightarrow \quad b^3+3 b=0 \Rightarrow b\left(b^2+3\right)=0$
$\Rightarrow \quad b=0, \pm \sqrt{3} i$