What is the answer to z^2=-3-4i ?

The answer to z^2=-3-4i is z=1-2i,z=-1+2i

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z^2=-3-4i

z^{2} = - 3 - 4 i

z = 1 - 2 i, z = - 1 + 2 i

Solution steps

z^{2} = - 3 - 4 i

Substitute

z = x + y i

(x + y i)^{2} = - 3 - 4 i

Expand

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

(x^{2} - y^{2}) + 2 i x y = - 3 - 4 i

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

[x^{2} - y^{2} = - 3 2 x y = - 4]

[x^{2} - y^{2} = - 3 2 x y = - 4] : (x = 1, x = - 1, y = - 2 y = 2)

Substitute back

z = x + y i

z = 1 - 2 i, z = - 1 + 2 i

(a + 6) + 2 bi = 6 - 5 i

(\frac{1 - i}{1 + i})^{100} = a + ib

∣ z - 2 + i ∣ = 2

z^{2} - 4 i z - 1 = 0

(2 x + y) + 6 i = 5 + (x + 4 y) i