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=-
,x1•x2=
.这个结论是法国数学家韦达最先发现并证明的,故把它称为“韦达定理”.利用此定理,可以不解方程就得出x1+x2和 x1•x2的值,进而求出相关的代数式的值.
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![数学公式](http://thumb.1010pic.com/pic5/latex/266131.png)
+…
的值.
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/pic5/latex/266131.png)
+…![](http://thumb.1010pic.com/pic5/latex/346878.png)
=-
[(
-
)+(
-
)+…+(
-
)]
=-
×(
-
)
=-
.
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.
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/pic5/latex/182213.png)
x1=
![](http://thumb.1010pic.com/pic5/latex/180190.png)
![](http://thumb.1010pic.com/pic5/latex/180191.png)
故有x1+x2=
![](http://thumb.1010pic.com/pic5/latex/180190.png)
![](http://thumb.1010pic.com/pic5/latex/180191.png)
![](http://thumb.1010pic.com/pic5/latex/124.png)
![](http://thumb.1010pic.com/pic5/latex/180190.png)
![](http://thumb.1010pic.com/pic5/latex/180191.png)
![](http://thumb.1010pic.com/pic5/latex/447.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/pic5/latex/346879.png)
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/656.png)
![](http://thumb.1010pic.com/pic5/latex/829.png)
∴
![](http://thumb.1010pic.com/pic5/latex/266131.png)
![](http://thumb.1010pic.com/pic5/latex/266132.png)
![](http://thumb.1010pic.com/pic5/latex/346878.png)
=-
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/8.png)
![](http://thumb.1010pic.com/pic5/latex/8.png)
![](http://thumb.1010pic.com/pic5/latex/96.png)
![](http://thumb.1010pic.com/pic5/latex/23406.png)
![](http://thumb.1010pic.com/pic5/latex/6477.png)
=-
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/6477.png)
=-
![](http://thumb.1010pic.com/pic5/latex/346880.png)
分析:(1)首先利用求根公式x=
![](http://thumb.1010pic.com/pic5/latex/182213.png)
![](http://thumb.1010pic.com/pic5/latex/124.png)
![](http://thumb.1010pic.com/pic5/latex/447.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/pic5/latex/346879.png)
![](http://thumb.1010pic.com/pic5/latex/13.png)
![](http://thumb.1010pic.com/pic5/latex/656.png)
![](http://thumb.1010pic.com/pic5/latex/829.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/pic5/latex/182213.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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