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

The answer to z^2+z=11+3i is z=-3.88327…-0.44335…i,z=2.88327…+0.44335…i

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z^2+z=11+3i

z^{2} + z = 11 + 3 i

z = - 3.88327 \dots - 0.44335 \dots i, z = 2.88327 \dots + 0.44335 \dots i

Solution steps

z^{2} + z = 11 + 3 i

Substitute

z = x + y i

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

Expand

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

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

[x^{2} - y^{2} + x = 11 y + 2 x y = 3] : (x = - 3.88327 \dots, x = 2.88327 \dots, y = - 0.44335 \dots y = 0.44335 \dots)

Substitute back

z = x + y i

z = - 3.88327 \dots - 0.44335 \dots i, z = 2.88327 \dots + 0.44335 \dots i

∣ 2 (a - i y) + i ∣ = 4

x^{2} = \frac{- 1 + i 3}{2}

x + 3 i = (1 + y i) (2 - 3 i)

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

z^{2} - 3 + 2 i = 0, z

What is the answer to z^2+z=11+3i ?
The answer to z^2+z=11+3i is z=-3.88327…-0.44335…i,z=2.88327…+0.44335…i