SOLUTION: Find all (real or nonreal) z satisfying (z - 3)^4 + (z - 5)^4 = -8.

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Question 1078278: Find all (real or nonreal) z satisfying
(z - 3)^4 + (z - 5)^4 = -8.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
%28z+-+3%29%5E4+%2B+%28z+-+5%29%5E4%22%22=%22%22-8

%28z+-+3%29%5E4+%2B+%28z+-+5%29%5E4%2B8%22%22=%22%220

The average of z-3 and z-5 is z-4, 
so let w = z-4, i.e., z = w+4

%28w%2B4+-+3%29%5E4+%2B+%28w%2B4+-+5%29%5E4%2B8%22%22=%22%220

%28w%2B1%29%5E4+%2B+%28w-1%29%5E4%2B8%22%22=%22%220

w%5E4%2B4w%5E3%2B6w%5E2%2B4w%2B1%2Bw%5E4-4w%5E3%2B6w%5E2-4w%2B1%2B8%22%22=%22%220

2w%5E4%2B12w%5E2%2B10%22%22=%22%220

Divide through by 2

w%5E4%2B6w%5E2%2B5%22%22=%22%220

%28w%5E2%2B5%29%28w%5E2%2B1%29%22%22=%22%220



Edwin