题目内容
阅读下列范例,按要求解答问题.例:已知实数a,b,c满足:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/0.png)
解:∵a+b+2c=1,∴a+b=1-2c,
设
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/1.png)
∵
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/2.png)
将①代入②得:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/3.png)
整理得:t2+(c2+2c+1)=0,即t2+(c+1)2=0,∴t=0,c=-1
将t,c的值同时代入①得:
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/4.png)
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/5.png)
以上解法是采用“均值换元”解决问题.一般地,若实数x,y满足x+y=m,则可设
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_ST/6.png)
已知实数a,b,c满足:a+b+c=6,a2+b2+c2=12,求a,b,c的值.
【答案】分析:从题中我们可以看出本题的关键是利用方程a+b+c=6得a+b=6-c,设
①
将①代入方程②a2+b2+c2=12,这就把三元的方程转化成二元的方程.求出未知数,就能正确的解出方程.
解答:解:∵a+b+c=6∴a+b=6-c,
设
①
∵a2+b2+c2=12②
∴![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_DA/2.png)
整理得:3c2-12c+4t2+12=0
配方得:3(c-2)2+4t2=0,
∴c=2,t=0
把c=2,t=0代入①得:a=2,b=2
所以,a=b=c=2.
点评:要用给出的换元技巧求方程的解,就要利用已给出的解题思路和方法,所以做这类题的关键是读懂给出的方法,这就要求学生平时一定要养成仔细读题的习惯.
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_DA/0.png)
将①代入方程②a2+b2+c2=12,这就把三元的方程转化成二元的方程.求出未知数,就能正确的解出方程.
解答:解:∵a+b+c=6∴a+b=6-c,
设
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_DA/1.png)
∵a2+b2+c2=12②
∴
![](http://thumb.1010pic.com/pic6/res/czsx/web/STSource/20131022163233910601092/SYS201310221632339106010022_DA/2.png)
整理得:3c2-12c+4t2+12=0
配方得:3(c-2)2+4t2=0,
∴c=2,t=0
把c=2,t=0代入①得:a=2,b=2
所以,a=b=c=2.
点评:要用给出的换元技巧求方程的解,就要利用已给出的解题思路和方法,所以做这类题的关键是读懂给出的方法,这就要求学生平时一定要养成仔细读题的习惯.
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目