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

• 1/2"
In summary, the conversation discusses a problem involving finding the value of x in a cubic equation. The poster attempted to solve it by substituting values and using a cube of x-1, but was not successful. Another poster suggests a solution by setting x equal to a fraction and solving for x. The conversation ends with the suggestion to show that this solution is equal to the given expression.
1/2"

## Homework Statement

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

## 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:
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

## 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.

## 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

## 1. What is the value of x in the equation x=2+2 2/3 +2 1/3?

The value of x in the equation x=2+2 2/3 +2 1/3 is 7.

## 2. How do you solve x=2+2 2/3 +2 1/3?

To solve x=2+2 2/3 +2 1/3, first add the whole numbers 2+2+2=6. Then, add the fractions 2/3+1/3=1. This gives us 6+1=7, so the solution is x=7.

## 3. Can this equation be simplified further?

Yes, the equation x=2+2 2/3 +2 1/3 can be simplified by converting the mixed numbers into improper fractions. This gives us x=2+8/3+7/3. Then, we can add the fractions and simplify to get x=7.

## 4. What if the equation was x=2+2 1/2 +2 1/2?

The solution would still be the same, x=7, as the fractions 2 1/2 and 2 1/3 are equivalent. This is because 1/2 and 1/3 are both equal to 2/6, so the values of the fractions do not change when multiplied by 2.

## 5. Can you use a calculator to solve this equation?

Yes, you can use a calculator to solve x=2+2 2/3 +2 1/3. Simply enter the equation as it is written and the calculator will give you the solution, which is x=7.

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