方程x2+(m-2)x+5-m=0的两根都大于2,则m的取值范围是(  ) A.(-5,-4] B.(-∞,-4] C.(-∞,-2] D.(-∞,-5)∪(-5,-4]

问题描述:

方程x2+(m-2)x+5-m=0的两根都大于2,则m的取值范围是(  )
A. (-5,-4]
B. (-∞,-4]
C. (-∞,-2]
D. (-∞,-5)∪(-5,-4]

令f(x)=x2+(m-2)x+5-m,其对称轴方程为x=

2−m
2

 由已知方程x2+(m-2)x+5-m=0的两根都大于2,故有
2−m
2
>2
f(2)>0
△≥0

 即
2−m
2
>2
4+2m−4+5−m>0
(m−2) 2−4(5−m)≥0
解得-5<m≤-4
   m的取值范围是(-5,-4]
   故应选A.