Question

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

• A. $-\sqrt{2}$
• B. $\pm \sqrt{3}i$
• C. $i\sqrt{5}$
• D. $\sqrt{2}$

Answer 1

The correct option is B

$\pm \sqrt{3}i$

Explanation for correct answer:

Step-1: Solving quadratic equation by cross multiplication.

Given, quadratic equation is $x^2+bx-1=0,x^2+x+b=0$ have one root common.

Solving by cross multiplication method.

$\Rightarrow$$\frac{x^2}{b^2-\left(-1\right)}=-\frac{x}{b-\left(-1\right)}=\frac{1}{1-b}$

$\Rightarrow$ $x=-\frac{b^2+1}{b+1}.........\left(i\right)$

and $x^2=\frac{b^2+1}{1-b}.........\left(ii\right)$

Step-2: Solving equation $\left(i\right)\&\left(ii\right)$

$\Rightarrow$ ${\left(\frac{b^2+1}{b+1}\right)}^2=\frac{b^2+1}{1-b}$

$\Rightarrow$ $\left(b^2+1\right)=\frac{b+1}{1-b}$

$\Rightarrow$ $\left(b+1\right)=\left(b^2+1\right)\left(1-b\right)$

$\Rightarrow$$b^3+2b+b=0$

$\Rightarrow$ $b\left(b^2+3\right)=0$

$\Rightarrow$ $b=0,\pm \sqrt{3}i$

Hence, option $\left(B\right)$ is correct.