Find all real or no real z satisfying (z-3)^4+(z-5)^4=-8.

(z-3)^4+(z-5)^4=-8   expanding, we have

z^4-12 z^3+54 z^2-108 z+81 + z^4-20 z^3+150 z^2-500 z+625  = - 8        simplify

2 z^4-32 z^3+204 z^2-608 z+714   = 0 

This will factor as :

2 (z^2-8 z+17) (z^2-8 z+21) = 0

Using the quadratic formula to solve the first quadratic factor, we have :

z = 4 - i

z = 4 + i

And using the quadratic formula to solve the second quadratic factor, we have

z =  4 - i sqrt(5) 

z =  4 + i sqrt(5)

Solve for z:
(z-5)^4+(z-3)^4 = -8

Expand out terms of the left hand side:
2 z^4-32 z^3+204 z^2-608 z+706 = -8

Add 8 to both sides:
2 z^4-32 z^3+204 z^2-608 z+714 = 0

The left hand side factors into a product with three terms:
2 (z^2-8 z+17) (z^2-8 z+21) = 0

Divide both sides by 2:
(z^2-8 z+17) (z^2-8 z+21) = 0

Split into two equations:
z^2-8 z+17 = 0 or z^2-8 z+21 = 0

Subtract 17 from both sides:
z^2-8 z = -17 or z^2-8 z+21 = 0

Add 16 to both sides:
z^2-8 z+16 = -1 or z^2-8 z+21 = 0

Write the left hand side as a square:
(z-4)^2 = -1 or z^2-8 z+21 = 0

Take the square root of both sides:
z-4 = i or z-4 = -i or z^2-8 z+21 = 0

Add 4 to both sides:
z = 4+i or z-4 = -i or z^2-8 z+21 = 0

Add 4 to both sides:
z = 4+i or z = 4-i or z^2-8 z+21 = 0

Subtract 21 from both sides:
z = 4+i or z = 4-i or z^2-8 z = -21

Add 16 to both sides:
z = 4+i or z = 4-i or z^2-8 z+16 = -5

Write the left hand side as a square:
z = 4+i or z = 4-i or (z-4)^2 = -5

Take the square root of both sides:
z = 4+i or z = 4-i or z-4 = i sqrt(5) or z-4 = -i sqrt(5)

Add 4 to both sides:
z = 4+i or z = 4-i or z = 4+i sqrt(5) or z-4 = -i sqrt(5)

Add 4 to both sides:
Answer: |z = 4+i   or   z = 4-i   or   z = 4+i sqrt(5)   or   z = 4-i sqrt(5)

Another approach:

Let y = z - 4 then we have:  (y + 1)^4 + (y - 1)^4  = -8

Expanding the terms in parentheses and collecting like terms:

2(y^4 + 6y^2 + 1) = -8 or:

y^4 + 6y^2 + 5 = 0

This factors nicely:  (y^2 + 1)((y^2 + 5) = 0

y^2 = -1  → y = i and y = -i.  Hence:  z = 4 + i and z = 4 - i

y^2 = -5  → y = i*sqrt(5) and y = -i*sqrt(5). Hence z = 4 + i*sqrt(5) and z = 4 - i*sqrt(5)

Thanks, Alan.......your approach makes this much easier.......how did you happen to hit on that particular substitution????

Alan: That is nice, but sounds like "Voodoo Math"!!