What is the answer to z^2=33-56i ?

The answer to z^2=33-56i is z=7-4i,z=-7+4i

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z^2=33-56i

z^{2} = 33 - 56 i

z = 7 - 4 i, z = - 7 + 4 i

Solution steps

z^{2} = 33 - 56 i

Substitute

z = x + y i

(x + y i)^{2} = 33 - 56 i

Expand

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

(x^{2} - y^{2}) + 2 i x y = 33 - 56 i

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

[x^{2} - y^{2} = 33 2 x y = - 56]

[x^{2} - y^{2} = 33 2 x y = - 56] : (x = 7, x = - 7, y = - 4 y = 4)

Substitute back

z = x + y i

z = 7 - 4 i, z = - 7 + 4 i

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

\frac{5 - 2 i}{z} = 2 + i

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

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

3 (x + 5 i) = 7 y i - 6 + 8