已知x为整数,且2/(x+3)+2/(3-x)+(2x+18)/(x^2-9)为整数,求所有符合条件x值的和.

问题描述:

已知x为整数,且2/(x+3)+2/(3-x)+(2x+18)/(x^2-9)为整数,求所有符合条件x值的和.

原式化简得到:2/(x-3)
所以x可以取1,2,4,5

化简为(2x+6)/(x^2-9)=2(x+3)/(x-3)(x+3)=2/(x-3)
所以x-3=+-1,+-2
所以x为4,2,1,5,和为12

[2/(x+3)]+[2/(3-x)]+[(2x+18)/(x^2-9)]
=[2(x-3)/(x^2-9)]-[2(x+3)/(x^2-9)]+[(2x+18)/(x^2-9)]
=(2x-6-2x-6+2x+18)/(x^2-9)
=(2x+6)/(x^2-9)
=2(x+3)/[(x+3)(x-3)]
=2/(x-3)
2/(x-3)为整数
2能被x-3整除
x-3=2,x=5
x-3=1,x=4
x-3=-1,x=2
x-3=-2,x=1

先通分并化简:原式=2/x-3
且X为整数,故X的值可为1,2,4,5。