已知x1、x2為方程x2+3x+1=0的兩實根,則x12+8x2+20= .
【答案】
分析:設x
12+8x
2+20=a,由根與系數關系,得x
1+x
2=-3,x
1•x
2=1,由已知,得x
12+3x
1+1=0,即x
12=-3x
1-1,將x
12+8x
2+20=a的左邊降次,與x
1+x
2=-3聯立求x
1,x
2,代入x
1•x
2=1中,求a的值.
解答:解:由已知,得x
1+x
2=-3,x
1•x
2=1,
又∵x
12+3x
1+1=0,即x
12=-3x
1-1,
∴x
12+8x
2+20=-3x
1+8x
2+19(設為a),
與x
1+x
2=-3聯立,得x
1=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155233815218475/SYS201310221552338152184013_DA/0.png)
,x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155233815218475/SYS201310221552338152184013_DA/1.png)
,
代入x
1•x
2=1中,得-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155233815218475/SYS201310221552338152184013_DA/2.png)
•
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155233815218475/SYS201310221552338152184013_DA/3.png)
=1,
整理,得a
2-23a-19=0,
解得a=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155233815218475/SYS201310221552338152184013_DA/4.png)
.
故答案為:
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022155233815218475/SYS201310221552338152184013_DA/5.png)
.
點評:本題考查了一元二次方程的根與系數關系.靈活運用根與系數關系,方程的解的意義是解題的關鍵.