实数A,B满足A*A*A+B*B*B+3ab=1,求A+B的值
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实数A,B满足A*A*A+B*B*B+3ab=1,求A+B的值
实数A,B满足A*A*A+B*B*B+3ab=1,求A+B的值
实数A,B满足A*A*A+B*B*B+3ab=1,求A+B的值
A*A*A+B*B*B+3AB=1
(A+B)^3-3AB(A+B)+3AB=1
(A+B)^3-1-[3AB(A+B)-3AB]=0
(A+B-1)[(A+B)^2+(A+B)+1]-3AB(A+B-1)=0
(A+B-1)[(A+B)^2+(A+B)+1-3AB]=0
即A+B-1=0或(A+B)^2+(A+B)+1-3AB=0
下面推出后者:
(A+B)^2+(A+B)+1-3AB
=A^2+2AB+B^2+A+B+1-3AB
=A^2-AB+B^2+A+B+1
=1/2*(A-B)^2+(1/2*A^2+A+1/2)+(1/2*B^2+B+1/2)
=1/2*(A-B)^2+1/2*(A+1)^2+1/2*(B+1)^2
最后一条式子大于等于0,当且仅当:A=B=-1
所以A+B=1或A+B=-2,后者A=B=-1
-2 a=b=-1
1 a,b一个为0一个为1
A=B=1/2
A+B=1/4
或者A=B=-1
A+B=-2
或者A=0,B=1
则A+B=1
1或-2
A=B=1/2
A+B=1
或者A=B=-1
A+B=-2
或者A=0,B=1
则A+B=1
1或-2
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