If α and β are the roots of the quadratic equation x 2 + x + 1 = 0 , then the equation whose roots are α 2021 , β 2021 is given by ______

Step-by-step solution

For quadratic equation, x 2 + x + 1 = 0 ⇒ x = - 1 ± 1 - 4 2 = - 1 ± 3 i 2 ⇒ α = - 1 + 3 i 2 ,   β = - 1 - 3 i 2 By using cube roots of unity, we can say α = ω   &     β = ω 2 So, α 2021 = ω 2021 = ω 3 673 . ω 2 = ω 2 , as ω 3 = 1 β 2021 = ω 2 2021 = ω 3 1347 . ω = ω Therefore, roots of new equation will be ω ,   ω 2 , Therefore, equation is x 2 - ω + ω 2 x + &#9