设实数x,y满足x2+2xy-1=0,则x+y的取值范围是?

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设实数x,y满足x2+2xy-1=0,则x+y的取值范围是?
设实数x,y满足x2+2xy-1=0,则x+y的取值范围是?

设实数x,y满足x2+2xy-1=0,则x+y的取值范围是?
x^2+2xy+y^2=1+y^2
(x+y)^2=1+y^2
因为1+y^2>=1
所以(x+y)^2〉=1
所以x+y>=1或x+y

x2+2xy-1=0
x²+2xy+y²=1+y²
(x+y)²=1+y²>=1+0=1
所以
|x+y|>=1
x+y<=-1或x+y>=1
x+y的取值范围是(负无穷，-1)并(1，正无穷)

设x＋y＝k，代入x^2+2xy-1=0
则x^2－2kx＋1＝0,
△＝4k －4≥0,所以k≥1或k≤－1
即x+y的取值范围是:(-∞,-1]∪[1,+∞).

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