Find all xx such that [x6x6 = (x+1)6(x+1)6].

So far, I have found the real solution x= -1212, and the complex solution x = -−1‾‾‾√3−13.

Are there more, and if so what/how would be the most efficient find all the solutions to this problem? I am struggling to find the rest of the solutions.

The question latex was messed up, it is: Find all $x$ such that

\[x^6 = (x+1)^6.\]

according to your correction we simply have

x^6 = (x+1)^6

take 6th root of both sides, since we are taking an even root,
x = ±(x+1)
x = x+1 or x = -x-1
x = x+1 ---> no solution
or
x = -x-1
2x = -1
x = -1/2

check:
LS = (-1/2)^6 = 1/64
RS = (-1/2+ 1)^6
= (1/2)^6 = 1/64

x = -1/2

Yeah, that is the real solution. I also need to find the complex solution as well.

For the complex solutions, note that you want

x^6 - (x+1)^6 = 0

This is just the difference of two squares, so it factors as

(x^3 - (x+1)^3)(x^3 + (x+1)^3) = 0
The left factor becomes just a simple quadratic when expanded, as the x^3 terms go away. The right factor is the sum of two cubes, so it i

(x + (x+1))(x^2-x(x+1)+x(x+1))
The first part gives the real root, and the second becomes just another quadratic.

work it out, and you should come up with the complex roots shown here

http://www.wolframalpha.com/input/?i=x%5E6+%3D+(x%2B1)%5E6

oops. the cubes factor as

(x + (x+1))(x^2-x(x+1)+(x+1)^2)

To find all the solutions to the equation x^6 + x^6 = (x+1)^6 * (x+1)^6, you can follow these steps:

Step 1: Expand the equation
Expanding both sides, we get:
2x^6 = (x+1)^12

Step 2: Simplify the equation
Take the square root of both sides:
√(2x^6) = √((x+1)^12)

Simplifying further, we get:
√2 * √(x^6) = (x+1)^6

Step 3: Rewrite the equation
Rewriting square roots as exponents, we have:
(2^(1/2)) * x^3 = (x+1)^6

Step 4: Expand again
Expand the right side of the equation:
(2^(1/2)) * x^3 = (x+1)(x+1)(x+1)(x+1)(x+1)(x+1)

Step 5: Simplify the equation
Expand the right side using binomial expansion or multiplication:
(2^(1/2)) * x^3 = (x^2 + 2x + 1)(x^2 + 2x + 1)(x^2 + 2x + 1)

Simplifying further, we get:
(2^(1/2)) * x^3 = (x^2 + 2x + 1)^3

Step 6: Solve for x
Now you have a cubic equation, where the left side is (2^(1/2)) * x^3 and the right side is (x^2 + 2x + 1)^3. To find all the solutions, you can solve this equation using various methods like factoring, graphing, or numerical techniques.

By solving this equation, you will find all the solutions for x. It seems you have already found two solutions (x = -1212 and x = -√3 - 1/3). To find additional solutions, you may need to use algebraic or numerical methods.