What is the answer to z^2-8z+18=8i ?

The answer to z^2-8z+18=8i is z=2.23276…-2.26342…i,z=5.76723…+2.26342…i

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z^2-8z+18=8i

z^{2} - 8 z + 18 = 8 i

z = 2.23276 \dots - 2.26342 \dots i, z = 5.76723 \dots + 2.26342 \dots i

Solution steps

z^{2} - 8 z + 18 = 8 i

Substitute

z = x + y i

(x + y i)^{2} - 8 (x + y i) + 18 = 8 i

Expand

(x + y i)^{2} - 8 (x + y i) + 18 : (x^{2} - y^{2} - 8 x + 18) + i (- 8 y + 2 x y)

(x^{2} - y^{2} - 8 x + 18) + i (- 8 y + 2 x y) = 8 i

Rewrite

8 i

in standard complex form:

0 + 8 i

(x^{2} - y^{2} - 8 x + 18) + i (- 8 y + 2 x y) = 0 + 8 i

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

[x^{2} - y^{2} - 8 x + 18 = 0 - 8 y + 2 x y = 8]

[x^{2} - y^{2} - 8 x + 18 = 0 - 8 y + 2 x y = 8] : (x = 2.23276 \dots, x = 5.76723 \dots, y = - 2.26342 \dots y = 2.26342 \dots)

Substitute back

z = x + y i

z = 2.23276 \dots - 2.26342 \dots i, z = 5.76723 \dots + 2.26342 \dots i

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

z^{2} + z = 5 + 5 i

z, z - \frac{1}{z} = 5 + 4 i

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

∣ z - i ∣ = 3

What is the answer to z^2-8z+18=8i ?
The answer to z^2-8z+18=8i is z=2.23276…-2.26342…i,z=5.76723…+2.26342…i