What is the answer to z^2=5-12i ?

The answer to z^2=5-12i is z=3-2i,z=-3+2i

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z^2=5-12i

z^{2} = 5 - 12 i

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

Solution steps

z^{2} = 5 - 12 i

Substitute

z = x + y i

(x + y i)^{2} = 5 - 12 i

Expand

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

(x^{2} - y^{2}) + 2 i x y = 5 - 12 i

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

[x^{2} - y^{2} = 5 2 x y = - 12]

[x^{2} - y^{2} = 5 2 x y = - 12] : (x = 3, x = - 3, y = - 2 y = 2)

Substitute back

z = x + y i

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

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

z^{2} - (5 + 4 i) z + 20 i = 0

4 (x - y i) = 8 + 2 y i + 10 i

(1 + 2 i) x + (3 - 5 i) y = 1 - 3 i

r (t) = t^{3} i + t^{2} j