已知X + X^-1 =3,求 X^1/2 + X^-1/2, X^1/2 - X^-1/2的值

来源：百度知道 编辑：UC知道 时间：2024/05/21 21:45:23
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(X^1/2 + X^-1/2)^2
=X+2+X^-1
=3+2
=5
所以X^1/2 + X^-1/2=±√5

(X^1/2-X^-1/2)^2
=X-2+X^-1
=3-2
=1
所以X^1/2-X^-1/2=±1

(X^1/2 + X^-1/2)^2=X + X^-1 +2=5
(X^1/2 + X^-1/2)=genhao5
(X^1/2 - X^-1/2)^2=X + X^-1 -2=1
X^1/2 - X^-1/2=+-1

X^1/2 + X^-1/2
＝±√(X^1/2 + X^-1/2)^2
=±√(X + 1/x+2(X^1/2*X^-1/2））
＝±√5

X^1/2 - X^-1/2
＝±√(X^1/2 － X^-1/2)^2
=±√(X + 1/x－2(X^1/2*X^-1/2））
1

X + X^-1 =3设X^1/2 为Y.
Y^2+(Y^2)^-1=3配方为（Y+Y^-1)^2-2=3
因为X^1/2 大于等于零（根号下吗！）
Y+Y^-1=根号5
X^1/2 + X^-1/2=根号5
同理:Y^2+(Y^2)^-1=3配方为（Y-Y^-1)^2+2=3
Y-Y^-1=1
X^1/2 - X^-1/2=±1

解：首先，由X + X^-1 =3知道X有两个解大于零，
且关于直线Y=X对称，那么
X^1/2 + X^-1/2的值大于零；
而X^1/2 - X^-1/2则可正可负；
(X^1/2 + X^-1/2)^2=X + (X^-1)+2=3+2=5,
所以X^1/2 + X^-1/2=5^1/2；
(X^1/2 - X^-1/2)^2=X + (X^-1)-2=3-2=1,
所以X^1/2 -X^-1/2=1或-1。