The question latex was messed up, it is: Find all $x$ such that
\[x^6 = (x+1)^6.\]
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.
5 answers
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
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
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)
(x + (x+1))(x^2-x(x+1)+(x+1)^2)