We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
805
1
avatar+644 

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

 Jan 25, 2018
 #1
avatar+101090 
+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

 Jan 25, 2018

17 Online Users

avatar