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

If α is a common root of x2+bx-1=0 and x2+x+b=0, we get α2+bα−1=0,α2+α+b=0.⇒bα−α−1−b=0,⇒α(b−1)=b+1⇒α=b+1b−1As α2+bα-1=0, we getb+1b−12+bb+1b−11−0⇒(b+1)2+b(b+1)(b−1)−(b−1)2=0⇒4b+bb2−1=0⇒b2+3=0⇒b=±3i

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

If α is a common root of x2+bx-1=0 and x2+x+b=0, we get α2+bα−1=0,α2+α+b=0.⇒bα−α−1−b=0,⇒α(b−1)=b+1⇒α=b+1b−1As α2+bα-1=0, we getb+1b−12+bb+1b−11−0⇒(b+1)2+b(b+1)(b−1)−(b−1)2=0⇒4b+bb2−1=0⇒b2+3=0⇒b=±3i

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A value of $b$ for which the equations
$\begin{array}{l} x^{2} + b x - 1 = 0 \\ x^{2} + x + b = 0 \end{array}$ have one root in common is

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$- \sqrt{2}$

$i \sqrt{5}$

$i \sqrt{3}$

$\sqrt{2}$

answer is C.

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Detailed Solution

If $α$ is a common root of $x^{2} + b x - 1 = 0$ and $x^{2} + x + b = 0,$ we get

$\begin{array}{l} α^{2} + b α - 1 = 0, α^{2} + α + b = 0 . \\ \Rightarrow b α - α - 1 - b = 0, \Rightarrow α (b - 1) = b + 1 \\ \Rightarrow α = \frac{b + 1}{b - 1} \end{array}$

As $α^{2} + b α - 1 = 0$ , we get

$\begin{array}{l} {(\frac{b + 1}{b - 1})}^{2} + b (\frac{b + 1}{b - 1}) 1 - 0 \\ \Rightarrow (b + 1)^{2} + b (b + 1) (b - 1) - (b - 1)^{2} = 0 \\ \Rightarrow 4 b + b (b^{2} - 1) = 0 \Rightarrow b^{2} + 3 = 0 \\ \Rightarrow b = \pm \sqrt{3} i \end{array}$