已知实数a,b满足a的三次方+b的三次方+3ab=1,则a+b=?

Question

已知实数a,b满足a的三次方+b的三次方+3ab=1,则a+b=?

Answer 1

a^3+b^3+3ab=1
(a+b)^3+3ab=1+3ab(a+b)
(a+b)^3-1=3ab(a+b-1)
(a+b-1)[(a+b)^2+(a+b)+1]=3ab(a+b-1)
(a+b-1)[(a+b)^2+(a+b)-3ab+1]=0
(a+b-1)[1/2(a+1)^2+1/2(b+1)^2+1/2(a-b)^2]=0
a+b=1,或a=b=-1
故a+b=1或a+b=-2