x^4=1/3(x^2-2/27)

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Solution for x^4=1/3(x^2-2/27) equation:


x in (-oo:+oo)

x^4 = (1/3)*(x^2-(2/27)) // - (1/3)*(x^2-(2/27))

x^4-((1/3)*(x^2-(2/27))) = 0

(-1/3)*(x^2-2/27)+x^4 = 0

x^4-1/3*(x^2-2/27) = 0

x^4-1/3*(x^2-2/27) = 0

x^4-1/3*x^2+2/81 = 0

x^4-1/3*x^2+2/81 = 0

x^4+(-1/3)*x^2+2/81 = 0

x^4+(-1/3)*x^2+2/81 = 0 // * 0

{ 1, -1, 2, -2 }

1

x = 1

81*x^4-27*x^2+2 = 56

1

-1

x = -1

81*x^4-27*x^2+2 = 56

-1

2

x = 2

81*x^4-27*x^2+2 = 1190

2

-2

x = -2

81*x^4-27*x^2+2 = 1190

-2

{ 1/3, -1/3, 1/9, -1/9, 1/27, -1/27, 1/81, -1/81, -1/3, 1/3, -1/9, 1/9, -1/27, 1/27, -1/81, 1/81, 2/3, -2/3, 2/9, -2/9, 2/27, -2/27, 2/81, -2/81, -2/3, 2/3, -2/9, 2/9, -2/27, 2/27, -2/81, 2/81 }

1/3

x

1/3

81*x^4-27*x^2+2 = 0

1/3

x-1/3

81*x^3+27*x^2-18*x-6

81*x^4-27*x^2+2

x-1/3

27*x^3-81*x^4

27*x^3-27*x^2+2

9*x^2-27*x^3

2-18*x^2

18*x^2-6*x

2-6*x

6*x-2

0

81*x^3+27*x^2-18*x-6 = 0

{ 1, -1, 2, -2, 3, -3, 6, -6 }

1

x = 1

81*x^3+27*x^2-18*x-6 = 84

1

-1

x = -1

81*x^3+27*x^2-18*x-6 = -42

-1

2

x = 2

81*x^3+27*x^2-18*x-6 = 714

2

-2

x = -2

81*x^3+27*x^2-18*x-6 = -510

-2

3

x = 3

81*x^3+27*x^2-18*x-6 = 2370

3

-3

x = -3

81*x^3+27*x^2-18*x-6 = -1896

-3

6

x = 6

81*x^3+27*x^2-18*x-6 = 18354

6

-6

x = -6

81*x^3+27*x^2-18*x-6 = -16422

-6

{ 1/3, -1/3, 1/9, -1/9, 1/27, -1/27, 1/81, -1/81, -1/3, 1/3, -1/9, 1/9, -1/27, 1/27, -1/81, 1/81, 2/3, -2/3, 2/9, -2/9, 2/27, -2/27, 2/81, -2/81, -2/3, 2/3, -2/9, 2/9, -2/27, 2/27, -2/81, 2/81, 3/3, -3/3, 3/9, -3/9, 3/27, -3/27, 3/81, -3/81, -3/3, 3/3, -3/9, 3/9, -3/27, 3/27, -3/81, 3/81, 6/3, -6/3, 6/9, -6/9, 6/27, -6/27, 6/81, -6/81, -6/3, 6/3, -6/9, 6/9, -6/27, 6/27, -6/81, 6/81 }

1/3

x

1/3

81*x^3+27*x^2-18*x-6 = -6

1/3

-1/3

x

-1/3

81*x^3+27*x^2-18*x-6 = 0

-1/3

x+1/3

81*x^2-18

81*x^3+27*x^2-18*x-6

x+1/3

-81*x^3-27*x^2

-18*x-6

18*x+6

0

81*x^2-18 = 0

DELTA = 0^2-(-18*4*81)

DELTA = 5832

DELTA > 0

x = (5832^(1/2)+0)/(2*81) or x = (0-5832^(1/2))/(2*81)

x = (2^(1/2))/3 or x = (-2^(1/2))/3

x in { (-2^(1/2))/3, (2^(1/2))/3, 1/3, -1/3}

(x-((-2^(1/2))/3))*(x-((2^(1/2))/3))*(x-1/3)*(x+1/3) = 0

(x-((-2^(1/2))/3))*(x-((2^(1/2))/3))*(x-1/3)*(x+1/3) = 0

( x+1/3 )

x+1/3 = 0 // - 1/3

x = -1/3

( x-((-2^(1/2))/3) )

x-((-2^(1/2))/3) = 0 // + (-2^(1/2))/3

x = (-2^(1/2))/3

( x-((2^(1/2))/3) )

x-((2^(1/2))/3) = 0 // + (2^(1/2))/3

x = (2^(1/2))/3

( x-1/3 )

x-1/3 = 0 // + 1/3

x = 1/3

x in { -1/3, (-2^(1/2))/3, (2^(1/2))/3, 1/3 }

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