Question

Fie x1, x2 solutiile ecuatiei x²+x+1=0. Să se calculeze:​

Answer 1

Răspuns:

x₁²⁰¹⁰+x₂²⁰¹⁰ = 1/2

Explicație pas cu pas:

x²+x+1=0 =>

x₁,₂ = (-1±√1-4)/2 = (-1±i√3)/2

x₁ = (-1-i√3)/2 ; x₂ = (-1+i√3)/2

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x₁² = [(-1-i√3)/2]² = (1+2i√3-3)/4 = -1/2·(i√3-1)

x₁⁴ = 1/4·(3i²+1-2i√3) = -1/2·(1+i√3)

x₁⁶ = 1/4·(1+2i√3-3) = -1/2·(1-i√3)

x₁²⁰¹⁰ = x₁⁶ =  -1/2·(1-i√3)

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x₂² =  [(-1+i√3)/2]² = (1-3-2i√3)/4 = -1/2·(1+i√3)

x₂⁴ = 1/4·(1+2i√3-3) = -1/2·(1-i√3)

x₂⁶ = 1/4·(1-2i√3+3) = 1/2·(2-i√3) = x₂²⁰¹⁰

x₁²⁰¹⁰+x₂²⁰¹⁰ = -1/2·(1-i√3) +  1/2·(2-i√3) =

= -1/2 + i√3/2 + 2/2 - i√3/2 = 2/2 - 1/2 = 1/2

x₁²⁰¹⁰+x₂²⁰¹⁰ = 1/2