Define f(z)=z^2+bz−1=0 and g(z)=z^2+z+b=0. If there exists α

define-f-z-z-2-bz-1-0-and-g-z-z-2-z-b-0-if-there-exists-a

Define f(z)=z^2+bz−1=0 and g(z)=z^2+z+b=0. If there exists α satisfying f(α)=g(α)=0, which of the following cannot be a value of b?

A. √3i

B. -√3i

C. 0

D. √3i/2

I had been asked this question in quiz.

Asked question is from Functions of a Complex Variable topic in chapter Complex Function Theory of Engineering Mathematics

Right choice is D. √3i/2

Explanation: α^2+bα−1=0 and α^2+α+b=0 ⇒ (b−1)α−1−b=0 ⇒ α=(b+1)/(b-1)

⇒ (b+1)^2/(b-1)^2+(b+1)/(b-1)+b=0 ⇒ b=√3i, -√3i, 0.