用配方法化二次型为标准型f(X1,x2,x3)=2x1x2-4x1x3+2x2^2-2x2x3

题目

用配方法化二次型为标准型f(X1,x2,x3)=2x1x2-4x1x3+2x2^2-2x2x3
我化的对吗：（x1+x2)^2+(X2-x3)^2-2(x1+x3)^2+x1^2+x3^2

答案

f= (2x2^2+2x1x2-2x2x3)-4x1x3
= 2(x2+x1/2-x3/2)^2 - (1/2)x1^2 -(1/2)x3^2-3x1x3
= 2(x2+x1/2-x3/2)^2 - (1/2)(x1+3x3)^2 +4x3^2