What is the answer to z^2-z=5-5i ?

The answer to z^2-z=5-5i is z=-2+i,z=3-i

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z^2-z=5-5i

z^{2} - z = 5 - 5 i

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

Solution steps

z^{2} - z = 5 - 5 i

Substitute

z = x + y i

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

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

[x^{2} - y^{2} - x = 5 - y + 2 x y = - 5]

[x^{2} - y^{2} - x = 5 - y + 2 x y = - 5] : (x = - 2, x = 3, y = 1 y = - 1)

Substitute back

z = x + y i

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

(1 + 3 i) z^{2} + (5 + 6 i) z + 7 - 6 i = 0

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

∣ z ∣^{2} + z - 3 i = 11

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

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