Question

If the equations x² + bx − 1 = 0 and x² + x + b = 0 have a common root different from −1, then |b| is equal to

(A)  2

(B)  3

(C)  √3

(D)  √2

Answer 1

Correct option (C)  √3

We have

x² + bx − 1 = 0 .... (1)

and x² + x + b = 0

⇒ b = −x − x²

Substituting the value of b in Eq. (1), we get

x² − x(x + x²) − 1 = 0

⇒ x³ + 1 = 0

⇒ (x + 1)(x² − x + 1) = 0

Therefore, x = −1. Also, 

x² − x + 1 = 0

⇒ x = −ω, ω²

where ω is the cube root of unity. Now,

b = −(x + x²)

⇒ b = - (-ω + ω²) = ω - ω² = i√3

⇒ |b| = √3