X+Y=10,X^3+Y^3=400,求X^2+Y^2

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X+Y=10,X^3+Y^3=400,求X^2+Y^2
X+Y=10,X^3+Y^3=400,求X^2+Y^2

X+Y=10,X^3+Y^3=400,求X^2+Y^2
(x+y)^3=x^3+3x^2 y+3xy^2+y^3=x^3+y^3+3xy(x+y)=400+3xy*10=1000
xy=20
(x+y)^2=x^2+y^2+2xy=100
x^2+y^2=60

x+y=10
x^2+y^2+2xy=100
x^2+y^2=100-2xy
x^3+y^3=(x+y)(x^2+y^2-xy)=400
则
10*(100-2xy-xy)=400
xy=20
则x^2+y^2=100-2*20=60

xy=50吧

400=x^3+y^3=(x+y)(x^2+y^2+xy)=(x+y)((x+y)^2-xy)=100(100-xy)
so xy=96
x^2+y^2=(x+y)^2-2xy=100-2*96=-92

因为
(x+y)^3=x^3+y^3+3xy(x+y)=10^3=1000
所以xy=20
之后有2种方法
1.x^3+y^3=(x+y)(x^2-xy+y^2)=400
所以x^2-xy+y^2=40 所以x^2+y^2=60
2.(x+y)^2=x^2+y^2+2xy=100
所以x^2+y^2=60

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