二元三次方程组x^3+y^3=10x^2y+xy^2=-3

Question

二元三次方程组
x^3+y^3=10
x^2y+xy^2=-3

Answer 1

令a=x+y,b=xy
x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)((x+y)^2-3xy)=a(a^2-3b)=a^3-3ab=10
yx^2+xy^2=xy(x+y)=ab=-3 代入上式得
a^3=1,解得a=1=x+y,所以b=-3=xy
将x=1-y代入b中得（1-y）y=-3,
解得y=0.5+0.5倍根号13 x=0.5-0.5倍根号13
或 x=0.5+0.5倍根号13 y=0.5-0.5倍根号13