Solving Quadratic Equations in the set of complex numbers

First I'd like to illustrate my solution to 6x^2 - 9 = 45, and then I'd like to get some

help solving 5x^2 +8 = x^2 - 8.

6x^2 - 9 = 45

6x^2 = 54

x^2 = 9

sqrt(x^2) = sqrt(9)

|x| = 3

x=-3, x=3

Now,

5x^2 + 8 = x^2 - 8

4x^2 = -16

x^2 = -4

sqrt(x^2) = sqrt(-4)

In my previous solution I used sqrt(x^2) = |x|, but in the set of complex numbers |a + bi| = sqrt(a^2 +b^2). The answer for an absolute value of a complex number does not have an imaginary unit in it. It does not make sense to me to write |x| = 2i.

I don't know how to justify that the solutions to sqrt(x^2) = sqrt(-4) are -2i and 2i