What is the answer to |i+z|^2-i-2=z ?

The answer to |i+z|^2-i-2=z is z=2-i,z=-1-i

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|i+z|^2-i-2=z

∣ i + z ∣^{2} - i - 2 = z

z = 2 - i, z = - 1 - i

Solution steps

∣ i + z ∣^{2} - i - 2 = z

Substitute

z = x + y i

∣ i + x + y i ∣^{2} - i - 2 = x + y i

Expand

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

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

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

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

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

Substitute back

z = x + y i

z = 2 - i, z = - 1 - i

a^{2} = i

i x^{2} + (1 - 2 i) x + 3 i - 1 = 0

x - 2 i = 2 + y i

z - \frac{1}{z} = 4 + 4 i

x^{2} + 2 i x - 1 - i = 0

What is the answer to |i+z|^2-i-2=z ?
The answer to |i+z|^2-i-2=z is z=2-i,z=-1-i