The 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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Aicharya · Accepted Answer

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

The 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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