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> 阅读理解题:一次数学兴趣小组的活动课上,师生有下面一段对话,请你阅读完后再解答下面问题:老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-8(x2-x)+12=0.学生甲:老师,-数学
阅读理解题:一次数学兴趣小组的活动课上,师生有下面一段对话,请你阅读完后再解答下面问题:老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-8(x2-x)+12=0.学生甲:老师,-数学
题目简介
阅读理解题:一次数学兴趣小组的活动课上,师生有下面一段对话,请你阅读完后再解答下面问题:老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-8(x2-x)+12=0.学生甲:老师,-数学
题目详情
阅读理解题:一次数学兴趣小组的活动课上,师生有下面一段对话,请你阅读完后再解答下面问题:
老师:同学们,今天我们来探索如下方程的解法:(x
2
-x)
2
-8(x
2
-x)+12=0.
学生甲:老师,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,用现有的知识无法解答.同学们再观察观察,看看这个方程有什么特点?
学生乙:我发现方程中x
2
-x是整体出现的,最好不要去括号!
老师:很好.如果我们把x
2
-x看成一个整体,用y来表示,那么原方程就变成y
2
-8y+12=0.
全体同学:咦,这不是我们学过的一元二次方程吗?
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的解是y
1
=6,y
2
=2,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,使用它最大的妙处在于降低了原方程的次数,这是一种很重要的转化方法.
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
(
x
x-1
)
2
-5(
x
x-1
)-6=0
.
题型:解答题
难度:中档
来源:青海
答案
设y=
class="stub"x
x-1
,
则原方程可变为y2-5y-6=0,
解得y1=6,y2=-1,
∴
class="stub"x
x-1
=6,
class="stub"x
x-1
=-1,
解得x=
class="stub"1
2
或
class="stub"6
5
,
经检验,都是原方程的根.
∴原方程的解为x=
class="stub"1
2
或
class="stub"6
5
.
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题目简介
阅读理解题:一次数学兴趣小组的活动课上,师生有下面一段对话,请你阅读完后再解答下面问题:老师:同学们,今天我们来探索如下方程的解法:(x2-x)2-8(x2-x)+12=0.学生甲:老师,-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)2-8(x2-x)+12=0.
学生甲:老师,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,用现有的知识无法解答.同学们再观察观察,看看这个方程有什么特点?
学生乙:我发现方程中x2-x是整体出现的,最好不要去括号!
老师:很好.如果我们把x2-x看成一个整体,用y来表示,那么原方程就变成y2-8y+12=0.
全体同学:咦,这不是我们学过的一元二次方程吗?
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的解是y1=6,y2=2,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,使用它最大的妙处在于降低了原方程的次数,这是一种很重要的转化方法.
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
则原方程可变为y2-5y-6=0,
解得y1=6,y2=-1,
∴
解得x=
经检验,都是原方程的根.
∴原方程的解为x=