题目内容
(2007•汕头)已知关于x的一元二次2x2-(2m2-1)x-m-4=0有一个实数根为![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_ST/0.png)
(1)求m的值;
(2)求已知方程所有不同的可能根的平方和.
【答案】分析:(1)由于
是方程的根,就直接代入方程得到关于m的方程,求出m的值,进而求出方程的形式;
(2)利用根与系数的关系就可求解第二问中的问题.
解答:解:(1)把x=
代入方程得:3m2+m-2=0,
解得m1=
,m2=-1;
(2)当m=
时,方程是2x2+
x-
=0,
设方程的两根是x1,x2,
则x1+x2=-
,
x1•x2=
,
则x12+x22=(x1+x2)2-2x1•x2=
;
当m=-1时方程是2x2-x-3=0,
设它的解是x3,x4,
则x3+x4=
,
x3•x4=-
,
∴x32+x42=(x3+x4)2-2x3x4=
+3=
,
∴x12+x22+x12+x22=
+
=![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/15.png)
点评:本题主要考查了根与系数的关系,解题时要灵活应用它把所求代数式化成可以直接求出的形式,在计算时一定要小心系数的符号.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/0.png)
(2)利用根与系数的关系就可求解第二问中的问题.
解答:解:(1)把x=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/1.png)
解得m1=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/2.png)
(2)当m=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/3.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/5.png)
设方程的两根是x1,x2,
则x1+x2=-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/6.png)
x1•x2=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/7.png)
则x12+x22=(x1+x2)2-2x1•x2=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/8.png)
当m=-1时方程是2x2-x-3=0,
设它的解是x3,x4,
则x3+x4=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/9.png)
x3•x4=-
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/10.png)
∴x32+x42=(x3+x4)2-2x3x4=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/11.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/12.png)
∴x12+x22+x12+x22=
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/13.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/14.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131021231638245868615/SYS201310212316382458686024_DA/15.png)
点评:本题主要考查了根与系数的关系,解题时要灵活应用它把所求代数式化成可以直接求出的形式,在计算时一定要小心系数的符号.
![](http://thumb2018.1010pic.com/images/loading.gif)
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