complex coefficients?

Question

can you guys help me with the last question of my homework?

solve the following. find all roots by completing the square.

x^2 - (i)x + 5 = 0

the i is an imaginary number.

this is the only problem i couldn't figure out. :( thanks!

Answer 1

(x-i/2)^2 = x^2 - ix + i^2/4
= x^2 - ix - 1/4

So x^2 - (i)x + 5 = (x-i/2)^2 + 5.25 = 0
(x-i/2)^2 = -21/4
(x-i/2) = +/- i*sqrt(21)/2
x = i/2 * (1 +/- sqrt(21))

Answer 2

x² - ix + 5 = 0
x² - ix = -5
x² - ix - 1/4 = -5 - 1/4
(x - i/2)² = -21/4
x - i/2 = ±i√21 / 2
x = (i ± i√21)/2