x^(1/2)-3x^(1/3)=3x^(1/6)-9

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Solution for x^(1/2)-3x^(1/3)=3x^(1/6)-9 equation: 


D( x )

x < 0

x < 0

x in <0:+oo)

x^(1/2)-(3*x^(1/3)) = 3*x^(1/6)-9 // - 3*x^(1/6)-9

x^(1/2)-(3*x^(1/3))-(3*x^(1/6))+9 = 0

x^(1/2)-3*x^(1/3)-3*x^(1/6)+9 = 0

t_1 = x^(1/6)

1*t_1^3-3*t_1^2-3*t_1^1+9 = 0

t_1^3-3*t_1^2-3*t_1+9 = 0

{ 1, -1, 3, -3, 9, -9 }

1

t_1 = 1

t_1^3-3*t_1^2-3*t_1+9 = 4

1

-1

t_1 = -1

t_1^3-3*t_1^2-3*t_1+9 = 8

-1

3

t_1 = 3

t_1^3-3*t_1^2-3*t_1+9 = 0

3

t_1-3

t_1^2-3

t_1^3-3*t_1^2-3*t_1+9

t_1-3

3*t_1^2-t_1^3

9-3*t_1

3*t_1-9

0

t_1^2-3 = 0

DELTA = 0^2-(-3*1*4)

DELTA = 12

DELTA > 0

t_1 = (12^(1/2)+0)/(1*2) or t_1 = (0-12^(1/2))/(1*2)

t_1 = 3^(1/2) or t_1 = -3^(1/2)

t_1 in { -3^(1/2), 3^(1/2), 3} 

t_1 = -3^(1/2)

x^(1/6)+3^(1/2) = 0

1*x^(1/6) = -3^(1/2) // : 1

x^(1/6) = -3^(1/2)

( -3^(1/2) < 0 i 1/6 in (0:1) ) => x naleu017Cy do O

t_1 = 3^(1/2)

x^(1/6)-3^(1/2) = 0

1*x^(1/6) = 3^(1/2) // : 1

x^(1/6) = 3^(1/2)

x^(1/6) = 3^(1/2) // ^ 6

x = 27

t_1 = 3

x^(1/6)-3 = 0

1*x^(1/6) = 3 // : 1

x^(1/6) = 3

x^(1/6) = 3 // ^ 6

x = 729

x in { 27, 729 }

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