题目内容
王老师要求学生进行编题.解题训练,其中小聪同学编的练习题是:设k=3,方程x2-3x+k=0的两个实数根是x1,x2,求
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_ST/0.png)
小明同学对这道题的解答过程是:
解:∵k=3,∴已知方程是x2-3x+3=0,
又∵x1+x2=3,x1•x2=3,
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_ST/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_ST/2.png)
即
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_ST/3.png)
(1)请你针对以上的练习题和解答的正误作出判断,再简述理由;
(2)请你只对小聪同学所编的练习题中的k另取一个适当的正整数,其他条件不变,改求
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_ST/4.png)
【答案】分析:(1)首先要判断方程的两根是否存在,即△≥0是否成立;
(2)取一个使△≥0的k的值后,根据
+
=
,代入求值.
解答:解:(1)错误;当k=3时,方程x2-3x+k=0即化为方程x2-3x+3=0,
△=(-3)2-4×1×3=-3<0,故方程无实根.
(2)要使方程x2-3x+k=0有两个实数根,
则△=(-3)2-4k≥0,即k≤
,故可取k=2,
则原方程变为x2-3x+2=0,
∵x1+x2=3,x1•x2=2,
+
=
=
=
.
点评:此类题目是中学阶段常规题目,解答时一定要先根据判别式△判断方程根的情况,再根据根与系数的关系解答,不能盲目计算.
(2)取一个使△≥0的k的值后,根据
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/0.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/2.png)
解答:解:(1)错误;当k=3时,方程x2-3x+k=0即化为方程x2-3x+3=0,
△=(-3)2-4×1×3=-3<0,故方程无实根.
(2)要使方程x2-3x+k=0有两个实数根,
则△=(-3)2-4k≥0,即k≤
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/3.png)
则原方程变为x2-3x+2=0,
∵x1+x2=3,x1•x2=2,
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/6.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/7.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131101193632887784349/SYS201311011936328877843019_DA/8.png)
点评:此类题目是中学阶段常规题目,解答时一定要先根据判别式△判断方程根的情况,再根据根与系数的关系解答,不能盲目计算.
![](http://thumb2018.1010pic.com/images/loading.gif)
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