Question

已知x为整数,且(2/x+3)+(2/3-x)+(2x+18/x²-9)为整数,求所有符合条件的x的值的和

Answer 1

2/(x+3)+2/(3-x)+(2x+18)/(x²-9)

=2/(x+3)-2/(x-3)+(2x+18)/(x²-9)

=[2(x-3)-2(x+3)+(2x+18)]/(x²-9)

=(2x+6)/[(x+3)(x-3)]

=2/(x-3)为正数,

又x为正数

∴x-3=2或者x-3=-2或x-2=1,或x-3=-1

∴x=5,1,3,2

所有符合条件的x的值的和为

5+1+3+2=11