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

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Solution

If $a_{1} x^{2} + b_{1} x + c_{1} = 0$ and $a_{2} x^{2} + b_{2} x + c_{2} = 0$
have a common real root, then
$\Rightarrow {(a_{1} c_{2} - a_{2} c_{1})}^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})$
$x^{2} + b x - 1 = 0, x^{2} + x + b = 0$ have a common root
$\Rightarrow {(1 + b)}^{2} = (b^{2} + 1) (1 - b)$
$\Rightarrow b^{2} + 2 b + 1 = b^{2} - b^{3} + 1 - b$
$\Rightarrow b^{3} + 3 b = 0$
$∴ b (b^{2} + 3) = 0$
$\Rightarrow b = 0, \pm \sqrt{3} i$

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