A complex variable z can be expressed as x+iy, where x and y are real numbers and i is the imaginary square root of -1. A complex variable can be represented as a point (x,y) where the y axis is the imaginary zone. All real numbers lie along the x axis, while complex numbers are represented as points in the x-y plane.
Complex solutions often arise as the roots or zeroes of a polynomial, for example, x^2+x+1=0 has no real solutions, but it has two complex solutions: x^2+x+1/4+3/4=0; (x+1/2)^2=-3/4; x+1/2=sqrt(-3/4)=isqrt(3)/2, so x=-1/2+isqrt(3)/2 and -1/2-isqrt(3)/2 are solutions of the quadratic x^2+x+1=0. x^2+x+1 factorises into:
(x+1/2-isqrt(3)/2)(x+1/2+isqrt(3)/2)