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Question: Answered & Verified by Expert
If $\mathrm{x}=2+2^{1 / 3}+2^{2 / 3}$, then what is the value of $x^{3}-6 x^{2}+6 x ?$
MathematicsQuadratic EquationNDANDA 2009 (Phase 2)
Options:
  • A 1
  • B 2
  • C -2
  • D 3
Solution:
1055 Upvotes Verified Answer
The correct answer is: 2
Let $\begin{aligned} x=2+2^{1 / 3}+2^{2 / 3} \\ &=2+2^{1 / 3}+\left(2^{1 / 3}\right)^{2}=2+2^{1 / 3}\left(1+2^{1 / 3}\right) \\ & \Rightarrow x-2=2^{1 / 3}\left(1+2^{1 / 3}\right) \end{aligned}$
On cubing both sides, we get
$x^{3}-8-6 x^{2}+12 x=2\left(1+2+3.2^{1 / 3}+3.2^{2 / 3}\right)$
$\Rightarrow x^{3}-6 x^{2}+6 x=14+6.2^{1 / 3}+6.2^{2 / 3}-6 x$
$\quad=14+6.2^{1 / 3}+6.2^{2 / 3}-6\left(2+2^{1 / 3}+2^{2 / 3}\right)=2$

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