Question

If the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have a common root different from $-1$, then $|b|$ is equal to:

• A. $\sqrt{2}$
• B. $3$
• C. $2$
• D. $\sqrt{3}$

Answer 1

The correct option is D $\sqrt{3}$

$x^2+bx-1=0$ ..........................(1)

and $x^2+x+b=0$....................(2)

Subtracting this to get common root as $x=\frac{b+1}{b-1}$

Substituting the common root in equation (2)

${\left(\frac{b+1}{b-1}\right)}^2+\left(\frac{b+1}{b-1}\right)+b=0$

${\left(\frac{b+1}{b-1}\right)}^2+\left(\frac{b+1}{b-1}\right)+b=0$

$(b+1{)}^2+(b+1\left)\right(b-1)+b(b-1{)}^2=0$

$b^2+2b+1+b^2-1+b(b^2+1-2b)=0$

$b^3+3b=0$

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

$0$ is not possible.

$|b|=\sqrt{3}$