I guess you mean a pair of pairs, as we expected to have two solutions that were normally real numbers. Yeah it's one of those times when the veil lifts and you find out you'd not seen the imaginary part until now. I guess enough has been said about completeness by other commentators, but there's no obvious answer as to why adding just one number solves your problem. Why won't it simply create new problems once you use those new complex numbers as coefficients in an expression? Surprisingly it's all we need.