What is the answer to iz+|z|+3i=1 ?

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

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iz+|z|+3i=1

i z + ∣ z ∣ + 3 i = 1

z = - 3 + 4 i

Solution steps

i z + ∣ z ∣ + 3 i = 1

Substitute

z = x + y i

i (x + y i) + ∣ (x + y i) ∣ + 3 i = 1

Simplify

i (x + y i) + ∣ x + y i ∣ + 3 i : (- y + x^{2} + y^{2}) + i (3 + x)

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

Rewrite

1

in standard complex form:

1 + 0 i

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

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

[- y + x^{2} + y^{2} = 1 3 + x = 0]

[- y + x^{2} + y^{2} = 1 3 + x = 0] : (x = - 3, y = 4)

Substitute back

z = x + y i

z = - 3 + 4 i

i (q) = 3 - q + q^{2}

x 3 + 2 x 2 + 5 s i x = 1

y, s i x^{2} + 8 y = y^{2}

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

- 3 + y i = x + 6 i