What is the answer to x^2-4x+4i=-1 ?

The answer to x^2-4x+4i=-1 is x=i,x=4-i

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x^2-4x+4i=-1

x^{2} - 4 x + 4 i = - 1

x = i, x = 4 - i

Solution steps

x^{2} - 4 x + 4 i = - 1

Substitute

x = a + bi

(a + bi)^{2} - 4 (a + bi) + 4 i = - 1

Expand

(a + bi)^{2} - 4 (a + bi) + 4 i : (a^{2} - b^{2} - 4 a) + i (4 - 4 b + 2 ab)

(a^{2} - b^{2} - 4 a) + i (4 - 4 b + 2 ab) = - 1

Rewrite

- 1

in standard complex form:

- 1 + 0 i

(a^{2} - b^{2} - 4 a) + i (4 - 4 b + 2 ab) = - 1 + 0 i

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

[a^{2} - b^{2} - 4 a = - 1 4 - 4 b + 2 ab = 0]

[a^{2} - b^{2} - 4 a = - 1 4 - 4 b + 2 ab = 0] : (a = 0, a = 4, b = 1 b = - 1)

Substitute back

x = a + bi

x = i, x = 4 - i

x + 3 y i = 4 - 5 i

(x - y) + 3 i = - 6 + y i

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

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

i (x) = - 2 x^{2} + 5 x + 1