What is the answer to z^2+|z|^2=50+3i ?

The answer to z^2+|z|^2=50+3i is z=-5-3/10 i,z=5+3/10 i

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z^2+|z|^2=50+3i

z^{2} + ∣ z ∣^{2} = 50 + 3 i

z = - 5 - \frac{3}{10} i, z = 5 + \frac{3}{10} i

Solution steps

z^{2} + ∣ z ∣^{2} = 50 + 3 i

Substitute

z = x + y i

(x + y i)^{2} + ∣ (x + y i) ∣^{2} = 50 + 3 i

Expand

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

2 x^{2} + 2 i x y = 50 + 3 i

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

[2 x^{2} = 50 2 x y = 3]

[2 x^{2} = 50 2 x y = 3] : (x = - 5, x = 5, y = - \frac{3}{10} y = \frac{3}{10})

Substitute back

z = x + y i

z = - 5 - \frac{3}{10} i, z = 5 + \frac{3}{10} 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

x 2 x - 4 y i = 10 - 5 i

What is the answer to z^2+|z|^2=50+3i ?
The answer to z^2+|z|^2=50+3i is z=-5-3/10 i,z=5+3/10 i