It is a difficult problem of pattern recognition based on clustering to solve the distance between a point in l-dimension space and a quadric surface.The latter has an important application in spacecraft laser-warning.The paper shows that the algebra equation stated above exits single root if the eigenvalues of A are all equal,and gives a recursion algorithm searching for the root.The upper or down boundaries of a Lagrangian multiplier are given if the eigenvalues of A take the same sign,which provides possibility and convenience for dichotomy solving Lagrangian multiplier.Resolving the problem will provide a new method for laser detecting and laser-warning.