關(guān)于x的方程x2+|x|-a2=0的所有實(shí)數(shù)根之和等于( )
A.-1
B.1
C.0
D.-a2
【答案】
分析:解方程時(shí)要分x≥0和x<0兩種情況解方程,然后再求所有實(shí)數(shù)根之和的值則可.
解答:解:方程x
2+|x|-a
2=0變形后為方程x
2+|x|=a
2若a=0時(shí),x也等于0,所有實(shí)數(shù)根之和也等于0.
若a不為零時(shí)當(dāng)x≥0時(shí)原方程x
2+|x|-a
2=0轉(zhuǎn)化為x
2+x-a
2=0
解得x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/0.png)
,x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/1.png)
,
∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/2.png)
>1,∴x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/3.png)
>0,x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/4.png)
<0(舍去).
當(dāng)x<0時(shí),原方程x
2+|x|-a
2=0轉(zhuǎn)化為x
2-x-a
2=0,
解得x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/5.png)
,x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/6.png)
,
∵
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/7.png)
>1,∴解得x
1=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/8.png)
>0(舍去),x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/9.png)
<0,
原方程得所有實(shí)數(shù)根之和等于
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/10.png)
+
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/11.png)
=0,
故選C.
點(diǎn)評:本題考查了運(yùn)用求根公式法解一元二次方程,本題易錯(cuò)易混點(diǎn):不會(huì)判斷
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/12.png)
與1的關(guān)系,因?yàn)閍
2>0,所以1+4a
2>1,
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155254863945726/SYS201310221552548639457000_DA/13.png)
也大于1;當(dāng)a=0時(shí),x為什么等于0,因?yàn)閤
2≥0,|x|≥0又因?yàn)閍=0,所以x=0.