What is the answer to z^2-4iz-6=0 ?

The answer to z^2-4iz-6=0 is z=sqrt(2)+2i,z=-sqrt(2)+2i

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z^2-4iz-6=0

z^{2} - 4 i z - 6 = 0

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

Solution steps

z^{2} - 4 i z - 6 = 0

Substitute

z = x + y i

(x + y i)^{2} - 4 i (x + y i) - 6 = 0

Expand

(x + y i)^{2} - 4 i (x + y i) - 6 : (x^{2} - y^{2} + 4 y - 6) + i (- 4 x + 2 x y)

(x^{2} - y^{2} + 4 y - 6) + i (- 4 x + 2 x y) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(x^{2} - y^{2} + 4 y - 6) + i (- 4 x + 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} + 4 y - 6 = 0 - 4 x + 2 x y = 0]

[x^{2} - y^{2} + 4 y - 6 = 0 - 4 x + 2 x y = 0] : (x = 2, x = - 2, y = 2 y = 2)

Substitute back

z = x + y i

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

z^{2} - (3 - 2 i) z + (5 - i) = 0

x + y i = (1 - i) (2 + 8 i)

2 (a + ib) + (3 + i) (a - ib) = 4 - i

1 = ai + b

(1 + 2 i) a + (3 - 5 i) b = 1 - 3 i

What is the answer to z^2-4iz-6=0 ?
The answer to z^2-4iz-6=0 is z=sqrt(2)+2i,z=-sqrt(2)+2i