What is the answer to z^2+2z=6-24i ?

The answer to z^2+2z=6-24i is z=3-3i,z=-5+3i

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z^2+2z=6-24i

z^{2} + 2 z = 6 - 24 i

z = 3 - 3 i, z = - 5 + 3 i

Solution steps

z^{2} + 2 z = 6 - 24 i

Substitute

z = x + y i

(x + y i)^{2} + 2 (x + y i) = 6 - 24 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) = 6 - 24 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 = 6 2 y + 2 x y = - 24]

[x^{2} - y^{2} + 2 x = 6 2 y + 2 x y = - 24] : (x = 3, x = - 5, y = - 3 y = 3)

Substitute back

z = x + y i

z = 3 - 3 i, z = - 5 + 3 i

x + y i = 2 - 3 i

z (z + 1) = 11 + 3 i

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

z^{2} + (3 + i) z + 4 + 3 i = 0

10 + 6 y i = 5 x + 24 i

What is the answer to z^2+2z=6-24i ?
The answer to z^2+2z=6-24i is z=3-3i,z=-5+3i