What is the answer to z^2+16=30i ?

The answer to z^2+16=30i is z=3+5i,z=-3-5i

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z^2+16=30i

z^{2} + 16 = 30 i

z = 3 + 5 i, z = - 3 - 5 i

Solution steps

z^{2} + 16 = 30 i

Substitute

z = x + y i

(x + y i)^{2} + 16 = 30 i

Expand

(x + y i)^{2} + 16 : (x^{2} - y^{2} + 16) + 2 i x y

(x^{2} - y^{2} + 16) + 2 i x y = 30 i

Rewrite

30 i

in standard complex form:

0 + 30 i

(x^{2} - y^{2} + 16) + 2 i x y = 0 + 30 i

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

[x^{2} - y^{2} + 16 = 0 2 x y = 30]

[x^{2} - y^{2} + 16 = 0 2 x y = 30] : (x = 3, x = - 3, y = 5 y = - 5)

Substitute back

z = x + y i

z = 3 + 5 i, z = - 3 - 5 i

- 2 i e z^{2} = - 1 + i, z^{1} = 3

(i) x^{2} - 16 = 0, (ii) 6 x^{2} - 24 = 0

z^{2} + 3 z + 2 i = 0

(2 i + 1)^{2} = (m + ni) (2 - i)

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