已知x+y=9，xy=20，求y+1x+1+x+1y+1

网友回答

∵x+y=9，xy=20，
∴原式=(y+1)
======以下答案可供参考======
供参考答案1:
y+1 /x+1 + x+1/ y+1
=(y+1)^2/(x+1)(y+1)+(x+1)^2/(x+1)(y+1)
=(y^2+2y+1+x^2+2x+1)/(xy+x+y+1)
=[(x^+y)^2-2xy+2(x+y)+2]/(xy+x+y+1)
=(81-40+18+2)/(20+9+1)
=61/30
供参考答案2:
x+y=9, （1）
xy=20 （2）
（1）²-（2）*4
（x-y)^2=1
x-y=1, (3)
x-y=-1 (4)
由（1）和（3）解得
x=5,y=4
y+1 /x+1 + x+1/ y+1
=5/6+6/5
=(25+36)/30
=61/30
由（1）和（4）解得
x=4,y=5
y+1 /x+1 + x+1/ y+1
=5/6+6/5
=(25+36)/30
=61/30
∴y+1 /x+1 + x+1/ y+1=61/30
供参考答案3:
(x+y)^2=x^2+2xy+y^2=81,xy=20,x^2+y^2=81-40=41,y+1/x+1+x+1/y+1=(y^2+2y+1+x^2+2x+1)/(xy+x+y+1)=61/30
供参考答案4:
y+1 /x+1 + x+1/ y+1
=(y+1)²/(x+1)(y+1)+(x+1)²/(x+1)(y+1)
=[(y+1)²+(x+1)²]/(x+1)(y+1)
=(x²+y²+2x+2y+2)/(xy+x+y+1)
=[(x+y)²-2xy+2(x+y)+2]/[xy+(x+y)+1]
=(81-40+18+2)/(20+9+1)
=61/30
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