In this paper, we investigate quadratic prediction problems of f ðYÞ ¼ Y0HY with H satisfying HX ¼ 0 in finite populations under the linear model Y ¼ Xb þ e, e Nð0; s2VÞ. We mainly aim at proposing two notions of optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor, and studying the structure of their explicit representations. In addition, we apply the main results of this paper to some special cases and finally derive a conclusion, which is not only mathematically interesting but also practically significant.Finite population; Optimal invariant quadratic unbiased predictor; Optimal invariant quadratic biased predictor; MINQU estimator; MRE estimator