What is the answer to z^2+z-2=z^2+zi+2 ?

The answer to z^2+z-2=z^2+zi+2 is z=2+2i

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z^2+z-2=z^2+zi+2

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

z = 2 + 2 i

Solution steps

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

Substitute

z = x + y i

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

Expand

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

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

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

[x^{2} - y^{2} + x - 2 = x^{2} - y^{2} - y + 2 y + 2 x y = x + 2 x y] : (x = 2, y = 2)

Substitute back

z = x + y i

z = 2 + 2 i

x, 9 x - 7 i = 3 (3 x - 7 u)

c = d i

z^{2} + 6 i z - 1 = 0

6 + \frac{1}{x} = 6, x^{2} + x^{- 2}

i (x) = 5 x - x^{2}

What is the answer to z^2+z-2=z^2+zi+2 ?
The answer to z^2+z-2=z^2+zi+2 is z=2+2i