x^3-1/x^3=76

Solution for x^3-1/x^3=76 equation:

D( x )

x^3 = 0

x^3 = 0

x^3 = 0

1*x^3 = 0 // : 1

x^3 = 0

x = 0

x in (-oo:0) U (0:+oo)

x^3-(1/(x^3)) = 76 // - 76

x^3-(1/(x^3))-76 = 0

x^3-x^-3-76 = 0

t_1 = x^3

1*t_1^1-1*t_1^-1-76 = 0

1*t_1^1-1*t_1^-1-76*t_1^0 = 0

(1*t_1^2-76*t_1^1-1*t_1^0)/(t_1^1) = 0 // * t_1^2

t_1^1*(1*t_1^2-76*t_1^1-1*t_1^0) = 0

t_1^1

t_1^2-76*t_1-1 = 0

t_1^2-76*t_1-1 = 0

DELTA = (-76)^2-(-1*1*4)

DELTA = 5780

DELTA > 0

t_1 = (5780^(1/2)+76)/(1*2) or t_1 = (76-5780^(1/2))/(1*2)

t_1 = (34*5^(1/2)+76)/2 or t_1 = (76-34*5^(1/2))/2

t_1 in { (76-34*5^(1/2))/2, (34*5^(1/2)+76)/2} 

t_1 = (76-34*5^(1/2))/2

x^3-((76-34*5^(1/2))/2) = 0

1*x^3 = (76-34*5^(1/2))/2 // : 1

x^3 = (76-34*5^(1/2))/2

x^3 = (76-34*5^(1/2))/2 // ^ 1/3

x = ((76-34*5^(1/2))^(1/3))/(2^(1/3))

t_1 = (34*5^(1/2)+76)/2

x^3-((34*5^(1/2)+76)/2) = 0

1*x^3 = (34*5^(1/2)+76)/2 // : 1

x^3 = (34*5^(1/2)+76)/2

x^3 = (34*5^(1/2)+76)/2 // ^ 1/3

x = ((34*5^(1/2)+76)^(1/3))/(2^(1/3))

x in { ((76-34*5^(1/2))^(1/3))/(2^(1/3)), ((34*5^(1/2)+76)^(1/3))/(2^(1/3)) }