题目内容
先阅读并完成第(1)题,再利用其结论解决第(2)题.(1)已知一元二次方程ax2+bx+c=0(a≠0)的两个实根为x1,x2,则有x1+x2=-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_ST/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_ST/1.png)
请你证明这个定理.
(2)对于一切不小于2的自然数n,关于x的一元二次方程x2-(n+2)x-2n2=0的两个根记作an,bn(n≥2),
请求出
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_ST/2.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_ST/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_ST/4.png)
【答案】分析:(1)首先利用求根公式x=
求得该方程的两个实数根,然后再来求得x1+x2=-
,x1•x2=
;
(2)由根与系数的关系得an+bn=n+2,an•bn=-2n2,所以(an-2)(bn-2)=anbn-2(an+bn)+4=-2n2-2(n+2)+4=-2n(n+1),
则
=-
(
-
),然后代入即可求解.
解答:解:(1)根据求根公式x=
知,
x1=
,x2=
,
故有x1+x2=
+
=-
,x1•x2=
×
=
;
(2)∵根与系数的关系知,an+bn=n+2,an•bn=-2n2,
∴(an-2)(bn-2)=anbn-2(an+bn)+4=-2n2-2(n+2)+4=-2n(n+1),
∴
=-
(
-
),
∴![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/20.png)
+…![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/22.png)
=-
[(
-
)+(
-
)+…+(
-
)]
=-
×(
-
)
=-
.
点评:本题考查了根与系数的关系.在证明韦达定理时,借用了求根公式x=
.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/2.png)
(2)由根与系数的关系得an+bn=n+2,an•bn=-2n2,所以(an-2)(bn-2)=anbn-2(an+bn)+4=-2n2-2(n+2)+4=-2n(n+1),
则
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/6.png)
解答:解:(1)根据求根公式x=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/7.png)
x1=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/8.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/9.png)
故有x1+x2=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/10.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/11.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/12.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/13.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/14.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/15.png)
(2)∵根与系数的关系知,an+bn=n+2,an•bn=-2n2,
∴(an-2)(bn-2)=anbn-2(an+bn)+4=-2n2-2(n+2)+4=-2n(n+1),
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/16.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/17.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/18.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/19.png)
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/20.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/21.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/22.png)
=-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/23.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/24.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/25.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/26.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/27.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/28.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/29.png)
=-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/30.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/31.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/32.png)
=-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/33.png)
点评:本题考查了根与系数的关系.在证明韦达定理时,借用了求根公式x=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022165058428434608/SYS201310221650584284346013_DA/34.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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