+647 Find all (real or nonreal) z satisfying
(z - 3)^4 + (z - 5)^4 = -8
+128475 ( 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
