Question

设集合A＝{x^2-4x+3=0},B={x|x^2-ax+a-1=0},C＝{x|x^2-mx+1},若A∪B=A,A∩C=C,求实数a,m的值

Answer 1

1)A={x|x^2-4x+3<0},
1<x<3,
2.1)x^2-2x+a<=0
(x-1)^2+a-1<=0;,a-1<=0,->a<=1;
2.2)x^2-2[a+7]x+5<=0
(x-(a+7))^2+5-(a+7)^2<=0;
5-(a+7)^2<=0;|a+7|>=√5，上面知道,
a<=1,a>=√5-7,√5-7<=a<=1,
A属于B,->x^2-2x+a<=0，x^2-2[a+7]x+5<=0中，
x在(1,3)范围内都取得到
1)x^2-2x+a<=0中x在(1,3）时，
(x-1)^2+a-1<=0,(x-1)^2∈(0,4)->a<=-3,
2)x^2-2[a+7]x+5<=0,x在(1,3）时,
是(x-(a+7))^2+5-(a+7)^2<=0,√5<a+7<8,
√5∈（1，3）,x=√5时，a=√5-7;
x=1,a=-4;x=√5,a=√5-7;x=3,x=-14/3;(过程省略)
a∈(√5-7,-4)
1）中a∈（-∞,-3），
终上所述：a∈(√5-7,-4),