What is the answer to z^2=8+6i ?

The answer to z^2=8+6i is z=3+i,z=-3-i

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z^2=8+6i

z^{2} = 8 + 6 i

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

Solution steps

z^{2} = 8 + 6 i

Substitute

z = x + y i

(x + y i)^{2} = 8 + 6 i

Expand

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

(x^{2} - y^{2}) + 2 i x y = 8 + 6 i

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

[x^{2} - y^{2} = 8 2 x y = 6]

[x^{2} - y^{2} = 8 2 x y = 6] : (x = 3, x = - 3, y = 1 y = - 1)

Substitute back

z = x + y i

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

(2 x - 3 y - 5) + i (x + 2 y + 1) = 0

5 x + 12 y i = - 10 + 4 i

x^{2} + 3 x + 3 - i = 0

x + y i = 0

\frac{m}{1 + i} + n = - m - \frac{1}{i}