# The value of $b$ for which the eqns., $x^2+bx-1=0,\:x^2+x+b=0$ have one common root is

$\begin{array}{1 1} \sqrt 2 \\ - \sqrt 2 \\ \sqrt{3 i} \\ \sqrt {5 i} \end{array}$

Given: $x^2+bx-1=0\:and\:x^2+x+b=0$ have one common root.
Let the common root be $\alpha$
$\Rightarrow\:\alpha^2+b\alpha-1=0 \: and\:\alpha^2+\alpha+b=0$
$\Rightarrow\:\large\frac{\alpha^2}{-1-b^2}=\frac{-\alpha}{-1-b}=\frac{1}{b-1}$
$\Rightarrow\:\alpha=\frac{b+1}{b-1}$,$\alpha=-\large\frac{1+b^2}{1+b}$$\:and\:\alpha^2=\frac{b^2+1}{1-b}$
$\Rightarrow\:(1+b)^2=(1+b^2)(1-b)$
$\Rightarrow\:b^2=-3$ or $b=\sqrt 3i$