Question

解方程：X^2+（X/X+1)^2=3(换元法）

Answer 1

令y=x+1 ,则原式化成 (y-1)^2 + (1-1/y)^2 = 3

把平方展开y^2 - 2y + 1 + 1- 2/y + (1/y)^2 = 3

整理得 [y^2+2+(1/y)^2] - 2(y+1/y) = 3

(y+1/y)^2 - 2(y+1/y) - 3 = 0

再令z=y+1/y 带入,得 z^2 - 2z - 3 = 0

z=3 或者 z= -1

y+1/y= -1没有实数根,y+1/y=3 解得y = (3+根号5)/2 或 y=(3-根号5)/2

最后解得x = [(3+根号5)/2] -1 或 x=[(3-根号5)/2]-1