关于x的方程1/x²+2x+10+1/x²+11x+10+1/x²-13x+10=0的解为

关于x的方程1/x²+2x+10+1/x²+11x+10+1/x²-13x+10=0的解为
关于x的方程1/x²+2x+10+1/x²+11x+10+1/x²-13x+10=0的解为

关于x的方程1/x²+2x+10+1/x²+11x+10+1/x²-13x+10=0的解为
写法容易引起歧义,是解1/(x²+2x+10)+1/(x²+11x+10)+1/(x²-13x+10) = 0吧?
∵1/(x²+2x+10)+1/(x²+11x+10)+1/(x²-13x+10) = 0,
∴1/(x+2+10/x)+1/(x+11+10/x)+1/(x-13+10/x) = x/(x²+2x+10)+x/(x²+11x+10)+x/(x²-13x+10) = 0.
换元y = x+10/x,可知1/(y+2)+1/(y+11)+1/(y-13) = 0.
∴3y²-147 = (y+11)(y-13)+(y+2)(y-13)+(y+2)(y+11) = 0,即有y² = 49.
解得y = ±7,代回x+10/x = y解得x = ±2,±5.
易验证它们都不是增根,因此原方程的解为x = ±2,±5.