Question

已知A={x|x²+(p+2)x+p=0,x∈R, p∈R}.

(1)若A∩{正實(shí)數(shù)}=,求p的取值范圍;

(2)若A∩{正實(shí)數(shù)},求p的取值范圍.

Answer 1

解：Δ=(p+2)²-4×p=(p-1)(p-4).

(1)∵A∩{正實(shí)數(shù)}=,

∴方程x²+(p+2)x+p=0無(wú)實(shí)數(shù)解或有非正實(shí)數(shù)解,于是Δ＜0,                        ①

或                                                      ②

解①得1＜p＜4;

解②得0≤p≤1或p≥4.

綜合①②知p≥0.

(2)方法一:由A∩{正實(shí)根},可知A集合中元素可能情況如下:

①兩正根;②一正根,一負(fù)根;③一零根,一正根;等價(jià)于

①或②x₁x₂＜0或③

由①②③知p＜0.

方法二:對(duì)于問(wèn)題(2)可轉(zhuǎn)化為在Δ≥0前提下A∩{正實(shí)數(shù)}與A∩{正實(shí)數(shù)}=是對(duì)立的,即在Δ≥0,即p≥4或p≤1情況下,{p|p≥0}的補(bǔ)集為{p|p＜0}.