NORMALIZED CUBIC EQUATION ROOTS

NORMALIZED CUBIC EQUATION ROOTS

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       a z 3 x + b z 2 + c z + d = 0        x = 4 m 3 n 2          a = 1        b = 1        m = 1 9 - c 3        n = c 3 - d - 2 27        c = - 3 m + 1 3        d = - m - n - 1 27

Normalized Cubic Equation The single real root is blue Analytical expressions for the curves are developed at cubic_exp.xml from the expressions reported at Weisstein   wikipedia and by   Knaust   The real and imaginary components of the derived expressions were found to be exact fits.

For   4 m 3 n 2 < 0 ,   z = 1 + 1 - 4 m 3 n 2 3 - - 1 + 1 - 4 m 3 n 2 3 - 1 3

For   0 < 4 m 3 n 2 < 1 ,   z = 1 + 1 - 4 m 3 n 2 3 + + 1 + 1 - 4 m 3 n 2 3 - 1 3

Exact agreement for the first real root when all roots are real. z = n 2 3 4 m 3 n 2 3 2 cos 0 π 6 + arctan 4 m 3 n 2 - 1 3 - 1 3

The imaginary components of the complex roots are red color coded curves. For   4 m 3 n 2 < 0 ,   z i = ± 3 2 1 + 1 - 4 m 3 n 2 3 - - 1 + 1 - 4 m 3 n 2 3

For   0 < 4 m 3 n 2 < 1 ,   z i = ± 3 2 1 + 1 - 4 m 3 n 2 3 + 1 - 1 - 4 m 3 n 2 3

The real component of the complex roots is color coded orange. For   4 m 3 n 2 < 0 ,   z r = 1 2 1 + 1 - 4 m 3 n 2 3 - - 1 + 1 - 4 m 3 n 2 3

The real component of the complex roots is color coded orange. For   0 < 4 m 3 n 2 < 1 ,   z r = 1 2 1 + 1 - 4 m 3 n 2 3 + + 1 + 1 - 4 m 3 n 2 3

fuchsia

cyan

magenta

pink

green z = n 2 3 4 m 3 n 2 3 cos 0 π 6 + arctan 4 m 3 n 2 - 1 3 - 1 3 ± n 2 3 4 m 3 n 2 3 3 sin 0 π 6 + arctan 4 m 3 n 2 - 1 3

maroon For an angle of pi/7 the solution is transcendental but for pi/60 the solution is only irrational

2 cos 2 π / 7 = 7 27 1 + 3 3 2 1 / 3 2 sin π - 2 3 atan 27 2 - 1 3

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