If k is one of the roots of the equation x(x + 1) + 1 = 0, then what is its other root?

Concept:

For a quadratic equation ax2 + bx + c = 0

The sum of the roots = \(\rm-{b\over a}\)

The product of the roots = \(\rm{c\over a}\)

Calculation:

Let the other root be β  

Given equation is x(x + 1) + 1 = 0

⇒ x² + x + 1 = 0

a = 1, b = 1 and c = 1

As k is the root of the equation

⇒ k2 + k + 1 = 0

⇒ k2 = -1 - k     .....(i)

The sum of the roots = \(-{1\over1}\) = -1

⇒ β + k = -1

⇒ β = -1 - k      .....(ii)

From equation (i) and (ii), we get

⇒ β = k²

∴ The other root = k²

Given equation is x(x + 1) + 1 = 0

Factor of (x2 + x + 1) = 0

\({\rm{x}} = {\rm{\;}}\frac{{ - 1{\rm{\;}} \pm {\rm{\;}}\sqrt {{1^2} - 4{\rm{\;}} \times 1{\rm{\;}} \times 1} }}{{2{\rm{\;}} \times 1}} = {\rm{\;}}\frac{{ - 1{\rm{\;}} \pm {\rm{i}}\sqrt 3 }}{2}\)

\(⇒ {\rm{x}} = {\rm{\;}}\frac{{ - 1 + {\rm{i}}\sqrt 3 }}{2}{\rm{\;or\;\;}}\frac{{ - 1 - {\rm{i}}\sqrt 3 }}{2}\)

⇒ x = ω or ω2

Consider k = ω 

∴ The other root = ω² = k2