Solving x=2+2 2/3 +2 1/3: A Puzzling Challenge

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Homework Statement

The problem is
If x= 2+2 ^2/3 +2 ^1/3
solve
x ³-6x ²+ 6x-2 =0

Homework Equations

The Attempt at a Solution

First i tried to substitute the values but its turning out to be really big( and i get quite scared when it turns up like this and so i didn't go any further this way)
then I tried to make a cube of x-1
Like this
x³ -1 -3x ²+ 3x-3x ²+3x-1+x ³=0
<==>(x-1)³+(x-1)³=x ³
<==>2(x-1)³=x ³
Well I can't get any further!
Happy if anyone helps!

 

[Mark44]

There's something screwy in your problem description. It's akin to saying "If x = 3, how many things are in a dozen?" There doesn't seem to be any connection between the two statements. Are we supposed to assume that x = 2 + 2^2/3 + 2^1/3 is a solution of the second equation?

 

[Mark44]

Homework Statement

The problem is
If x= 2+2 ^2/3 +2 ^1/3
solve
x ³-6x ²+ 6x-2 =0

Homework Equations

The Attempt at a Solution

First i tried to substitute the values but its turning out to be really big( and i get quite scared when it turns up like this and so i didn't go any further this way)
then I tried to make a cube of x-1

Why did you pick x - 1? I can't think of any good reason to do this.

Also, I have no idea of what you're trying to do in the work below.

Like this
x³ -1 -3x ²+ 3x-3x ²+3x-1+x ³=0
<==>(x-1)³+(x-1)³=x ³
<==>2(x-1)³=x ³
Well I can't get any further!
Happy if anyone helps!

 

[LCKurtz]

Homework Statement

The problem is
If x= 2+2 ^2/3 +2 ^1/3
solve
x ³-6x ²+ 6x-2 =0

I presume you mean "show", not "solve". You are trying to show that value of x is a root of the cubic.

Homework Equations

The Attempt at a Solution

First i tried to substitute the values but its turning out to be really big( and i get quite scared when it turns up like this and so i didn't go any further this way)
then I tried to make a cube of x-1
Like this
x³ -1 -3x ²+ 3x-3x ²+3x-1+x ³=0
<==>(x-1)³+(x-1)³=x ³
<==>2(x-1)³=x ³
Well I can't get any further!
Happy if anyone helps!

I think you just quit too soon.
$$2 = \left (\frac {x}{x-1}\right)^3$$
$$\frac x {x-1} = 2^{\frac 1 3}$$

Solve for x:

$$x = \frac{2^{\frac 1 3}}{2^{\frac 1 3}-1}$$

Now all you have to do is show this is equal to 2+2 ^2/3 +2 ^1/3