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. $-\sqrt2$
• B. $-i\sqrt3$
• C. $i\sqrt5$
• D. $\sqrt2$

Answer 1

Answer: (B)

$-i\sqrt3$

Answer 2

The correct answer is B:$-i\sqrt3$
If $a_1x^2+b_1x+c_1=0 \ and\ a_2x^2+b_2x+c_2=0$
have a common real root, then
$\Rightarrow\space20mm(a_1 c_2-a_2 c_1)^2=(b_1 c_2-b_2 c_1)(a_1 b_2-a_2 b_1)$ $$\begin{array}{l} x^2 + bx - 1 = 0 \\ x^2 + x + b = 0 \end{array} \quad \Bigg\{ \text{have a common root}$$$\Rightarrow\space25mm (1+b^2)=(b^2+1)(1-b)$$\Rightarrow\space20mmb^2+2b+1=b^2-b^3+1-b$$\Rightarrow\space25mm b^3+3b=0$$\therefore\space25mm b(b^2+3)=0 \Rightarrow b=0,\pm \sqrt3 \ i$