A value of b for which the equations x2+bx−1=0 and x2+x+b=0 have one root in common, is

# A value of b for which the equations ${\mathrm{x}}^{2}+\mathrm{bx}-1=0$ and ${\mathrm{x}}^{2}+\mathrm{x}+\mathrm{b}=0$ have one root in common, is

1. A

$-\sqrt{2}$

2. B

$-\mathrm{i}\sqrt{3}$

3. C

$\mathrm{i}\sqrt{5}$

4. D

$\sqrt{2}$

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### Solution:

We know that, if ${\mathrm{a}}_{1}{\mathrm{x}}^{2}+{\mathrm{b}}_{1}\mathrm{x}+{\mathrm{c}}_{1}=0$ and ${\mathrm{a}}_{2}{\mathrm{x}}^{2}+{\mathrm{b}}_{2}^{2}\mathrm{x}+{\mathrm{c}}_{2}=0$  have a common real root, then ${\left({\mathrm{a}}_{1}{\mathrm{c}}_{2}-{\mathrm{a}}_{2}{\mathrm{c}}_{1}\right)}^{2}=\left({\mathrm{b}}_{1}{\mathrm{c}}_{2}-{\mathrm{b}}_{2}{\mathrm{c}}_{1}\right)\left({\mathrm{a}}_{1}{\mathrm{b}}_{2}-{\mathrm{a}}_{2}{\mathrm{b}}_{1}\right)$ then ${\left({\mathrm{a}}_{1}{\mathrm{c}}_{2}-{\mathrm{a}}_{2}{\mathrm{c}}_{1}\right)}^{2}=\left({\mathrm{b}}_{1}{\mathrm{c}}_{2}-{\mathrm{b}}_{2}{\mathrm{c}}_{1}\right)\left({\mathrm{a}}_{1}{\mathrm{b}}_{2}-{\mathrm{a}}_{2}{\mathrm{b}}_{1}\right)$
$\begin{array}{l}⇒\left(1+\mathrm{b}{\right)}^{2}=\left({\mathrm{b}}^{2}+1\right)\left(1-\mathrm{b}\right)\\ ⇒{\mathrm{b}}^{2}+2\mathrm{b}+1={\mathrm{b}}^{2}-{\mathrm{b}}^{3}+1-\mathrm{b}\\ ⇒{\mathrm{b}}^{3}+3\mathrm{b}=0⇒\mathrm{b}\left({\mathrm{b}}^{2}+3\right)=0\\ ⇒\mathrm{b}=0,±\mathrm{i}\sqrt{3}\end{array}$  Register to Get Free Mock Test and Study Material

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