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

The answer to z^2-2z=3i is z=-0.44261…-1.03977…i,z=2.44261…+1.03977…i

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z^2-2z=3i

z^{2} - 2 z = 3 i

z = - 0.44261 \dots - 1.03977 \dots i, z = 2.44261 \dots + 1.03977 \dots i

Solution steps

z^{2} - 2 z = 3 i

Substitute

z = x + y i

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

Expand

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

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

Rewrite

3 i

in standard complex form:

0 + 3 i

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

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

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

[x^{2} - y^{2} - 2 x = 0 - 2 y + 2 x y = 3] : (x = - 0.44261 \dots, x = 2.44261 \dots, y = - 1.03977 \dots y = 1.03977 \dots)

Substitute back

z = x + y i

z = - 0.44261 \dots - 1.03977 \dots i, z = 2.44261 \dots + 1.03977 \dots i

i = 2 x^{2} + x - 2

(a - bi) (a + bi) + (a - bi) = 3 - i

(x - y) + 8 i = 6 + y i

2 p - q + i (7 i + 3) = 2 (2 i - q) - i (p + 3 q)

y - 5 = - i [x - (- 5)]

What is the answer to z^2-2z=3i ?
The answer to z^2-2z=3i is z=-0.44261…-1.03977…i,z=2.44261…+1.03977…i