已知一元二次方程x2+3x+1=0的兩根為x1和x2,那么(1+x1)(1+x2)的值為 .
【答案】
分析:利用一元二次方程根與系數(shù)的關(guān)系可知x
1+x
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022160039265814075/SYS201310221600392658140015_DA/0.png)
=-3,x
1•x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022160039265814075/SYS201310221600392658140015_DA/1.png)
=1,而(1+x
1)(1+x
2)=1+x
1+x
2+x
1•x
2,再把前面的值代入即可求出其值.
解答:解:∵一元二次方程x
2+3x+1=0的兩根為x
1和x
2,
∴x
1+x
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022160039265814075/SYS201310221600392658140015_DA/2.png)
=-3,x
1•x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022160039265814075/SYS201310221600392658140015_DA/3.png)
=1,
而(1+x
1)(1+x
2)
=1+x
1+x
2+x
1•x
2=1-3+1
=-1.
故填空答案-1.
點評:本題考查了一元二次方程根與系數(shù)的關(guān)系.一元二次方程ax
2+bx+c=0(a≠0)的根與系數(shù)的關(guān)系為:x
1+x
2=-
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022160039265814075/SYS201310221600392658140015_DA/4.png)
,x
1•x
2=
![](http://thumb.zyjl.cn/pic6/res/czsx/web/STSource/20131022160039265814075/SYS201310221600392658140015_DA/5.png)
.在形如x
2+px+q=0的方程中x
1+x
2=-p,x
1•x
2=q.