[cmantzioros]

In our exercises we are told to solve for x (element of a complex number)

1. x^2 - 6x + 25=0

The answer is x=3+4i or x=3-4i

Can anyone tell me how these answers were derived?

[CroCivic91]

In our exercises we are told to solve for x (element of a complex number)

1. x^2 - 6x + 25=0

The answer is x=3+4i or x=3-4i

Can anyone tell me how these answers were derived??

A more general problem than you have is this: a*x^2 + b*x + c = 0
In your case, a = 1, b = -6 and c = 25
There is a general formula for solving such equations, and it's "solutions" are given by these formulas:
x1 = (-b + square_root( b^2 - 4*a*c ) ) / (2*a)
x2 = (-b - square_root( b^2 - 4*a*c ) ) / (2*a)

In your example, you get:
x1 = (-(-6) + square_root( (-6)^2 - 4*1*25 ) ) / (2*1) = ( 6 + square_root( -64 ) ) / 2 = ( 6 + 8*i ) / 2 = 3 + 4i
x2 = ( 6 - 8i ) / 2 = 3 - 4i