已知非零实数x,y满足:x2+xy-2y2=0,求(x2+3y+y2)/(x2+y2)的值

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已知非零实数x,y满足:x2+xy-2y2=0,求(x2+3y+y2)/(x2+y2)的值

已知非零实数x,y满足:x2+xy-2y2=0,求(x2+3y+y2)/(x2+y2)的值
已知非零实数x,y满足:x2+xy-2y2=0,求(x2+3y+y2)/(x2+y2)的值

已知非零实数x,y满足:x2+xy-2y2=0,求(x2+3y+y2)/(x2+y2)的值
由x2+xy-2y2=0,
x2-y2+xy-y2=0,
(x+y)(x-y)+(x-y)y=0,
(x-y)(x+2y)=0.
得x-y=0或者x+2y=0.
1)当x-y=0,x=y.
(x2+3y+y2)/(x2+y2)=(x2+3x+x2)/(x2+x2)=(2x2+3x)/(2x2)=1+3/2x;
2)当x+2y=0,x=-2y.
(x2+3y+y2)/(x2+y2)=(4y2+3y+y2)/(4y2+y2)=(5y2+3y)/(5y2)=1+3/5y.

x2+xy-2y2=0
(x+2y)(x-y)=0
x=-2y或x=y
(x2+3y+y2)/(x2+y2)=1+3y/(x2+y2)
x=-2y时,x^2=4y^2,1+3y/(x2+y2)=1+3/(5y)
x=y时,x^2=y^2,1+3y/(x2+y2)=1+3/(2y)

x² - xy - 2y² = 0
方程两边同除y²得:
(x/y)² -(x/y) - 2 = 0
(x/y + 1)(x/y -2) = 0
x/y = -1 或 x/y = 2
(x2+xy+y2)/(x2+y2)=1+xy/(x2+y2)=1+1/((x/y)+(y/x))代入x/y = -1 或 x/y = ...

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x² - xy - 2y² = 0
方程两边同除y²得:
(x/y)² -(x/y) - 2 = 0
(x/y + 1)(x/y -2) = 0
x/y = -1 或 x/y = 2
(x2+xy+y2)/(x2+y2)=1+xy/(x2+y2)=1+1/((x/y)+(y/x))代入x/y = -1 或 x/y = 2即为所求

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