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

The answer to |z|-z=2+1i is z=-3/4-i

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|z|-z=2+1i

∣ z ∣ - z = 2 + 1 i

z = - \frac{3}{4} - i

Solution steps

∣ z ∣ - z = 2 + 1 i

Substitute

z = x + y i

∣ (x + y i) ∣ - (x + y i) = 2 + 1 i

Expand

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

[x^{2} + y^{2} - x = 2 - y = 1] : (x = - \frac{3}{4}, y = - 1)

Substitute back

z = x + y i

z = - \frac{3}{4} - i

4 x - i y = 8 + 7 i

3 x + 6 i = 27 + y i

z, z^{2} = (x + i y)^{2}

∣ 2 i + 6 j + 2 ik - 4 jk ∣ = 5

(x + y i) + (4 - 7 i) = 3 - 4 i