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Let us say you have a general quadratic x2 +bx+c where b2 -4c<0 & the quadratic's roots are α & β
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then we know x2 +bx+c≡ (x-α )(x-β )
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therefore, if expand (x-α )(x-β ) we get x2-(α+β)x+αβ now if you let α=λ+iμ then β=λ-iμ we can do this because the coefficients of quadratic are always real in fp1 therefore complex roots occur in pairs.
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α+β=2λ and αβ=λ2+μ2
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now we get to the important form: x2 +bx+c≡x2-2λx+(λ2+μ2)
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