If x = 2 + 23 + 23 , prove that x3-6x2 + 6x-2-0

1.	If x = 2 + 23 + 23 , prove that x3-6x2 + 6x-2-0
Answer» Given that x = 2 + 2^(2/3) + 2^(1/3) so x - 2 = 2^(2/3) + 2^(1/3) =(x-2)^3 = [2^(2/3)+2^(1/3)]^3 using (a-b)^3= a^3 - b^3 - 3ab(a-b) and (a+b)^3= a^3 + b^3 + 3ab(a+b) formulae. or x^3 - 8 - 6x(x-2) = 2^2 + 2^1 + 3*[2^{(2/3)+(1/3)}[2^(2/3)+2^(1/3) or x^3 - 8 - 6x^2 + 12x = 4 + 2 + 6(x-2) or x^3 - 8 -6x^2 + 12x = 6 + 6x - 12 or x^3 - 6x^2 +6x = 2

Answer»

Given that x = 2 + 2^(2/3) + 2^(1/3)

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

=(x-2)^3 = [2^(2/3)+2^(1/3)]^3

using (a-b)^3= a^3 - b^3 - 3ab(a-b)

and (a+b)^3= a^3 + b^3 + 3ab(a+b) formulae.

or x^3 - 8 - 6x(x-2) = 2^2 + 2^1 + 3*[2^{(2/3)+(1/3)}[2^(2/3)+2^(1/3)

or x^3 - 8 - 6x^2 + 12x = 4 + 2 + 6(x-2)

or x^3 - 8 -6x^2 + 12x = 6 + 6x - 12

or x^3 - 6x^2 +6x = 2

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