What is the answer to |z|^2+z=11+3i ?

The answer to |z|^2+z=11+3i is z=1+3i,z=-2+3i

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|z|^2+z=11+3i

∣ z ∣^{2} + z = 11 + 3 i

z = 1 + 3 i, z = - 2 + 3 i

Solution steps

∣ z ∣^{2} + z = 11 + 3 i

Substitute

z = x + y i

∣ (x + y i) ∣^{2} + x + y i = 11 + 3 i

Simplify

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

(x^{2} + y^{2} + x) + i y = 11 + 3 i

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

[x^{2} + y^{2} + x = 11 y = 3]

[x^{2} + y^{2} + x = 11 y = 3] : (x = 1, x = - 2, y = 3 y = 3)

Substitute back

z = x + y i

z = 1 + 3 i, z = - 2 + 3 i

z^{2} + ∣ z ∣^{2} = 50 + 3 i

(1 + i) z^{2} = - 1 + 7 i

5 + bi = (a + b i^{2})

z, (2 - i) z + 8 z^{2} = 0

(4 + i) x + (9 - 2 i) y = - 57 + 24 i

What is the answer to |z|^2+z=11+3i ?
The answer to |z|^2+z=11+3i is z=1+3i,z=-2+3i