What is the answer to z^2-5+12i=0 ?

The answer to z^2-5+12i=0 is z=3-2i,z=-3+2i

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z^2-5+12i=0

z^{2} - 5 + 12 i = 0

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

Solution steps

z^{2} - 5 + 12 i = 0

Substitute

z = x + y i

(x + y i)^{2} - 5 + 12 i = 0

Expand

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

(x^{2} - y^{2} - 5) + i (12 + 2 x y) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(x^{2} - y^{2} - 5) + i (12 + 2 x y) = 0 + 0 i

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

[x^{2} - y^{2} - 5 = 0 12 + 2 x y = 0]

[x^{2} - y^{2} - 5 = 0 12 + 2 x y = 0] : (x = 3, x = - 3, y = - 2 y = 2)

Substitute back

z = x + y i

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

x, y sin (y) - 2 - 5 x^{2} = 0

z^{2} + (3 - i) z + (2 + i) = 0

i \frac{1}{16} x + 6 y = 7

15 - 28 i = 3 l + (4 m) i

x^{2} = 1184 - 125.6637 i