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

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The discussion revolves around solving the equation x = 2 + 2 2/3 + 2 1/3 and determining if this value is a root of the cubic equation x^3 - 6x^2 + 6x - 2 = 0. Participants express confusion over the connection between the two statements and the complexity of substituting the values directly. One user attempts to manipulate the equation using the expression (x-1)^3 but struggles to progress further. Another contributor suggests that the goal is to show that the derived value of x satisfies the cubic equation, ultimately leading to a solution involving cube roots. The conversation highlights the challenges in solving polynomial equations and the importance of clarifying problem statements.
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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!
 
Last edited:
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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 + 22/3 + 21/3 is a solution of the second equation?
 
1/2" said:

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.
1/2" said:
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!
 
1/2" said:

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
 

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