+129845 z = a + bi i^2 = -1
( a + bi)^2 = 2i + 2
a^2 + 2abi + b^2i^2 = 2i + 2
a^2 - b^2 + 2ab i = 2i + 2
Which implies that
a^2 - b^2 = 2 (1)
2ab = 2 ⇒ ab = 1 ⇒ b = 1/a (2)
Sub (2) into (1)
a^2 - (1/a)^2 = 2 mutiply through by a^2
a^4 - 1 = 2a^2
a^4 - 2a^2 - 1 = 0 let a^2 = x ⇒ a = sqrt (x)
x^2 - 2x - 1 = 0
x^2 - 2x = 1 complete the square on x
x^2 - 2x + 1 = 1 + 1
(x - 1)^2 = 2
x - 1 = ±sqrt (2)
x = 1 + sqrt (2) or 1 - sqrt (2)
Only the first solution is valid
a = sqrt (x) = sqrt ( 1 + sqrt (2) )
b = 1 / sqrt ( 1 + sqrt (2) )
z = sqrt ( 1 + sqrt (2) ) + 1/ (sqrt ( 1 + sqrt (2)) i
