In which of the following equations they have roots α and β?

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Answer 1 · 2017-10-31T08:44:17

See below.

If #alpha,beta# are roots for a real second degree polynomial then must be complex conjugate. #alpha = bar(beta)#

but #a = e^(i (2pi)/7)# and

#alpha = e^((2 I pi)/7) + e^((4 i pi)/7) + e^(-(6 i pi)/7)# then

#beta = e^(-(2 I pi)/7) + e^(-(4 i pi)/7) + e^((6 i pi)/7)#

Now #alpha* beta = 2# and #alpha + beta = -1#

but

#(x-alpha)(x-beta) = x^2-(alpha+beta)x+alpha*beta# so #alpha, beta# are the roots for

#x^2+x+2=0#