Question

ÔĶÁÏÂÁнâÌâ¹ý³Ì£º
ÒÑÖª

x

x2+1

=

1

3

£¬Çó

x2

x2+1

µÄÖµ£®
½â£ºÓÉ

x

x2+1

=

1

3

£¬Öªx¡Ù0£¬ËùÒÔ

x2+1

x

=3£¬¼´x+

1

x

=3£®
¡à

x4+1

x2

=x²+

1

x2

=(x+

1

x

)2-2=3²-2=7£®
¡à

x2

x2+1

µÄֵΪ7µÄµ¹Êý£¬¼´

1

7

£®
ÒÔÉϽⷨÖÐÏȽ«ÒÑÖªµÈʽµÄÁ½±ß¡°È¡µ¹Êý¡±£¬È»ºóÇó³öÌØÇóʽ×Óµ¹ÊýµÄÖµ£¬ÎÒÃÇ°Ñ´ËÌâµÄÕâÖֽⷨ½Ð×ö¡°µ¹Êý·¨¡±£¬ÇëÄãÀûÓá°µ¹Êý·¨¡±½â¾öÏÂÃæÎÊÌ⣺
£¨1£©ÒÑÖª

x

x2-x+1

=7£¬Çó

x2

x4+x2+1

µÄÖµ£®
£¨2£©ÒÑÖª

xy

x+y

=2£¬

yz

y+z

=

4

3

£¬

zx

z+x

=

4

3

£¬Çó

xyz

xy+yz+zx

µÄÖµ£®

Answer 1

¿¼µã£º·ÖʽµÄ»¯¼òÇóÖµ

רÌ⣺ÔĶÁÐÍ

·ÖÎö£º£¨1£©ÒÑÖªµÈʽ±äÐÎÇó³öx+

1

x

µÄÖµ£¬Ô­Ê½±äÐκ󣬽«x+

1

x

µÄÖµ´úÈë¼ÆËã¼´¿É£»
£¨2£©ÒÑÖªÈýµÈʽ±äÐκóÏà¼ÓÇó³ö

1

x

+

1

y

+

1

z

µÄÖµ£¬Ô­Ê½±äÐκó´úÈë¼ÆËã¼´¿ÉÇó³öÖµ£®

½â´ð£º½â£º£¨1£©

x

x2-x+1

=

1

x+ 1 x -1

=7£¬¼´x+

1

x

=

8

7

£¬
Ôòԭʽ=

1

x2+ 1 x2 +1

=

1

(x+ 1 x )2-1

=

1

64 49 -1

=

49

15

£»
£¨2£©¸ù¾ÝÌâÒâµÃ£º

x+y

xy

=

1

x

+

1

y

=

1

2

£¬

y+z

yz

=

1

y

+

1

z

=

3

4

£¬

z+x

zx

=

1

x

+

1

z

=

3

4

£¬
¿ÉµÃ

1

x

+

1

y

+

1

z

=1£¬
Ôòԭʽ=

1

1 x + 1 y + 1 z

=1£®

µãÆÀ£º´ËÌ⿼²éÁË·ÖʽµÄ»¯¼òÇóÖµ£¬ÊìÁ·ÕÆÎÕÔËËã·¨ÔòÊǽⱾÌâµÄ¹Ø¼ü£®