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

The answer to |z|-z=1+2i is z= 3/2-2i

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|z|-z=1+2i

∣ z ∣ - z = 1 + 2 i

z = \frac{3}{2} - 2 i

Solution steps

∣ z ∣ - z = 1 + 2 i

Substitute

z = x + y i

∣ (x + y i) ∣ - (x + y i) = 1 + 2 i

Expand

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

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

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

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

[x^{2} + y^{2} - x = 1 - y = 2] : (x = \frac{3}{2}, y = - 2)

Substitute back

z = x + y i

z = \frac{3}{2} - 2 i

(x + i y) (2 + 3 i) = 1 - 5 i

∣ z - i ∣ = 2

z^{2} = - \frac{2}{3} i

W^{2} = 3 + 4 i

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