用换元法解方程x^2+1/(x^2)-9/2(x+1/x)+7=0,所设的辅助未知数y=?则原方程化为?

问题描述:

用换元法解方程x^2+1/(x^2)-9/2(x+1/x)+7=0,所设的辅助未知数y=?则原方程化为?

令,X+1/X=Y,
(X^2+1/X^2+2)-9/2(X+1/X)+5=0,
Y^2-9/2Y+5=0,
2Y^2-9Y+10=0,
Y1=5/2,Y2=2.
X+1/X=5/2,或X+1/X=2(方程无解,不合,舍去)
X1=1/2,X2=2,

y=1+1/x
y^2-9/2y+5=0

x^2+1/(x^2)-9/2*(x+1/x)+7=0(x+1/x)^2 -2 -9/2*(x+1/x)+7=0(x+1/x)^2 -9/2*(x+1/x)+5=0令y=x+1/x则 y^2 -9/2*y+5=02y^2-9y+10=0解得y=2 或y=5/2代入 y=x+1/xx+1/x=2x^2-2x+1=0(x-1)^2=0x=1x+1/x=5/22x^2-5x+1=0x=(...