使方程x^2-ax+9a=0 的根都為整數(shù)的實數(shù)a的所有徝之和為

數(shù)學人氣：520 ℃時間：2020-06-18 00:15:08

優(yōu)質(zhì)解答

設根是m和n
則m+n=a
mn=9a
所以mn=9m+9n
(n-9)m=9n
m=9n/(n-9)=(9n-81+81)/(n-9)=9(n-9)/(n-9)+81/(n-9)=9+81/(n-9)
m是整數(shù)
所以n-9是81的約數(shù)
所以n-9=1,-1,3,-3,9,-9,27,-27,81,-81
n=10,7,12,6,18,0,36,-18,90,-72
則相應的m=90,-63,36,-18,18,0,12,6,10,8
a=m+n
所以a=100,-54,48,-12,36,0,48,-12,100,-64
所以a有7個值
a=100,-54,48,-12,36,0,-64
所以a的所有徝之和=100-54+48-12+36-64=54