+0  
 
0
453
1
avatar+644 

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

waffles  Jan 25, 2018
 #1
avatar+89775 
+1

( z - 3)^4 + (z - 5)^4  = - 8

 

Let    a + 1  =  z - 3      and  a - 1  = z  - 5   

 

So we have

 

(a + 1)^4  +  ( a - 1)^4  = - 8

 

a^4 + 4a^3 + 6a^2 + 4a + 1  + a^4 - 4a^3 + 6a^2 - 4a + 1  =  - 8

 

2a^4 +  12a^2  + 2  =  - 8

 

2a^4  + 12a^2  + 10  =  0

 

a^4 + 6a^2  + 5  =  0        factor

 

(a^2 + 5) ( a^2 + 1)  = 0

 

Setting each factor to 0  and solving for a we have that

 

a = i√5 , -i√5, i , -i

 

So...

 

a =  z - 4  ⇒  a + 4  = z

 

z =   4 + i√5,  4 - i√5,  4 + i,   4 - i

 

 

cool cool cool

CPhill  Jan 25, 2018

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