What is the answer to z^2=-4i ?

The answer to z^2=-4i is z=sqrt(2)-sqrt(2)i,z=-sqrt(2)+sqrt(2)i

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z^2=-4i

z^{2} = - 4 i

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

Solution steps

z^{2} = - 4 i

Substitute

z = x + y i

(x + y i)^{2} = - 4 i

Expand

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

(x^{2} - y^{2}) + 2 i x y = - 4 i

Rewrite

- 4 i

in standard complex form:

0 - 4 i

(x^{2} - y^{2}) + 2 i x y = 0 - 4 i

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

[x^{2} - y^{2} = 0 2 x y = - 4]

[x^{2} - y^{2} = 0 2 x y = - 4] : (x = 2, x = - 2, y = - 2 y = 2)

Substitute back

z = x + y i

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

z^{2} + 2 - 4 i = 0

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

x^{2} + 2 i x - 2 = 0

z, z^{2} - (1 - 4 i) z - 4 i = 0

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

What is the answer to z^2=-4i ?
The answer to z^2=-4i is z=sqrt(2)-sqrt(2)i,z=-sqrt(2)+sqrt(2)i