Question

If x = 2 + 2^1/3+2^(-1/3).. prove that): 2x^3 - 12x^2 + 18x - 9 = 0...
help for it please
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Answer 1

Answer:

$2x^3-12x^2+18x-9=0$

Step-by-step explanation:

$Formula\:used:\\\\(a+b)^3=a^3+b^3+3ab(a+b)$

$Given:\\\\x=2+2^{\frac{1}{3}}+2^{\frac{-1}{3}}\\\\x-2=2^{\frac{1}{3}}+2^{\frac{-1}{3}}\\\\Raising\:both\:sides\:to\:power\:3\\\\(x-2)^3=[2^{\frac{1}{3}}+\frac{1}{2^{\frac{1}{3}}}]^3$

$x^3+(-2)^3+3x(-2)(x-2)=(2^{\frac{1}{3}})^3+(\frac{1}{2^{\frac{1}{3}}})^3+3(2^{\frac{1}{3}})(\frac{1}{2^{\frac{1}{3}}})(2^{\frac{1}{3}}+\frac{1}{2^{\frac{1}{3}}})\\\\x^3-8-6x^2+12x=2+\frac{1}{2}+3(2^{\frac{1}{3}}+\frac{1}{2^{\frac{1}{3}}})\\\\x^3-10-6x^2+12x=\frac{1}{2}+3(x-2)\\\\x^3-10-6x^2+12x=\frac{1}{2}+3x-6\\\\x^3-6x^2+9x-4=\frac{1}{2}\\\\2x^3-12x^2+18x-8=1\\\\2x^3-12x^2+18x-9=0$