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> 在一次数学兴趣小组的活动课上,师生有下面的一段对话,请你阅读完后再解答问题.老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-(x2-x)+12=0学生甲:老师,这个方程先去-九年级数学
在一次数学兴趣小组的活动课上,师生有下面的一段对话,请你阅读完后再解答问题.老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-(x2-x)+12=0学生甲:老师,这个方程先去-九年级数学
题目简介
在一次数学兴趣小组的活动课上,师生有下面的一段对话,请你阅读完后再解答问题.老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-(x2-x)+12=0学生甲:老师,这个方程先去-九年级数学
题目详情
在一次数学兴趣小组的活动课上,师生有下面的一段对话,请你阅读完后再解答问题.
老师:同学们,今天我们来探索如下方程的解法:
(x
2
-x)
2
-(x
2
-x)+12=0
学生甲:老师,这个方程先去括号,再合并同类项,行吗?
老师:这样,原方程可整理为x
4
-2x
3
-7x
2
+8x+12=0,次数变成了4次,用现有知识无法解答.同学们再观察观察,看看这个方程有什么特点?
学生乙:老师,我发现x
2
-x是整体出现的,最好不要去括号!
老师:很好,我们把x
2
-x看成一个整体,用y表示,即x
2
-x=y,那么原方程就变为y
2
+8y+12=0.
全体学生:(同学们都特别高兴)噢,这不是我们熟悉的一元二次方程吗?!
老师:大家真会观察和思考,太棒了!显然一元二次方程y
2
+8y+12=0的根是y
1
=6,y
2
=2,那么就有x
2
-x=6或x
2
-x=2.
学生丙:对啦,再解这两个方程,可得原方程的根x
1
=3,x
2
=-2,x
3
=2,x
4
=-1,嗬,有这么多根啊!
老师:同学们,通常我们把这种方法叫做换元法.在这里使用它的最大妙处在于降低了原方程的次数,这是一种重要的转化方法.
全体同学:OK,换元法真神奇!
现在,请你用换元法解下列分式方程:
题型:解答题
难度:中档
来源:河北省期末题
答案
解:设y=
,则原方程化为y2-5y-6=0
解关于y的一元二次方程得:y1=6,y2=-1
将y=6代入y=
中,x1=
将y=-1代入y=
中,x2=
将x1=
,x2=
代入原方程检验知,它们均为原方程的根。
所以方程
的解为x1=
,x2=
.
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题目简介
在一次数学兴趣小组的活动课上,师生有下面的一段对话,请你阅读完后再解答问题.老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-(x2-x)+12=0学生甲:老师,这个方程先去-九年级数学
题目详情
老师:同学们,今天我们来探索如下方程的解法:
(x2-x)2-(x2-x)+12=0
学生甲:老师,这个方程先去括号,再合并同类项,行吗?
老师:这样,原方程可整理为x4-2x3-7x2+8x+12=0,次数变成了4次,用现有知识无法解答.同学们再观察观察,看看这个方程有什么特点?
学生乙:老师,我发现x2-x是整体出现的,最好不要去括号!
老师:很好,我们把x2-x看成一个整体,用y表示,即x2-x=y,那么原方程就变为y2+8y+12=0.
全体学生:(同学们都特别高兴)噢,这不是我们熟悉的一元二次方程吗?!
老师:大家真会观察和思考,太棒了!显然一元二次方程y2+8y+12=0的根是y1=6,y2=2,那么就有x2-x=6或x2-x=2.
学生丙:对啦,再解这两个方程,可得原方程的根x1=3,x2=-2,x3=2,x4=-1,嗬,有这么多根啊!
老师:同学们,通常我们把这种方法叫做换元法.在这里使用它的最大妙处在于降低了原方程的次数,这是一种重要的转化方法.
全体同学:OK,换元法真神奇!
现在,请你用换元法解下列分式方程:
答案
解关于y的一元二次方程得:y1=6,y2=-1
将y=6代入y=
将y=-1代入y=
将x1=
所以方程