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

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

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

∣ z ∣^{2} + z - 3 i = 11

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

Solution steps

∣ z ∣^{2} + z - 3 i = 11

Substitute

z = x + y i

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

Simplify

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

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

Rewrite

11

in standard complex form:

11 + 0 i

(x^{2} + y^{2} + x) + i (- 3 + y) = 11 + 0 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 - 3 + y = 0]

[x^{2} + y^{2} + x = 11 - 3 + y = 0] : (x = 1, x = - 2, y = 3 y = 3)

Substitute back

z = x + y i

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

im (\frac{1}{x + y i}) = \frac{1}{2}

z^{2} = - \frac{4}{3} i

5 x^{2} + b = (5 x + 8 i) (5 x - 8 i)

x + (x - y) i = 4 i

a^{2} + b^{2} + 1 + 12 i = 6 (a + bi)

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