What is the answer to (z-i)/z =z+i ?

The answer to (z-i)/z =z+i is z=-0.36602…+0.36602…i,z=1.36602…-1.36602…i

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(z-i)/z =z+i

\frac{z - i}{z} = z + i

z = - 0.36602 \dots + 0.36602 \dots i, z = 1.36602 \dots - 1.36602 \dots i

Solution steps

\frac{z - i}{z} = z + i

Substitute

z = x + y i

\frac{x + y i - i}{x + y i} = x + y i + i

Multiply both sides by

x + y i

\frac{x + y i - i}{x + y i} (x + y i) = x (x + y i) + y i (x + y i) + i (x + y i)

Expand

x + i (- 1 + y) = (x^{2} - y^{2} - y) + i (x + 2 x y)

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[x = x^{2} - y^{2} - y - 1 + y = x + 2 x y]

[x = x^{2} - y^{2} - y - 1 + y = x + 2 x y] : (x = - 0.36602 \dots, x = 1.36602 \dots, y = 0.36602 \dots y = - 1.36602 \dots)

Substitute back

z = x + y i

z = - 0.36602 \dots + 0.36602 \dots i, z = 1.36602 \dots - 1.36602 \dots i

x + i y = (3 x + 4 i) (1 + x - i y)

z^{2} = i (∣ z ∣^{2} - 4)

(2 - 3 i) x + (3 - 2 i) y = 14 + 8 i

y - in y = x^{2} + 1

(a + b + 3) + (8 + b) i = 2 a + (3 a - b) i

What is the answer to (z-i)/z =z+i ?
The answer to (z-i)/z =z+i is z=-0.36602…+0.36602…i,z=1.36602…-1.36602…i