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分式方程和无理方程的增根问题是近几年中考以及竞赛命题的热点和难点,由于这类问题并不是把所有的条件都直接明了地告诉考生,而是把某些条件隐含在问题的结论或数学式子中,解答时,别说是考生,就是数学教育工作者也难以防范,现举例说明,供借鉴. 例1 方程2x/(x+1)-k/(x2+x)=x+1/x只有唯一解,求k. 同学们是这样解答的: 去分母,得2x2-k=(x+1)2, x2-2x-k-1=0. 因为方程只有唯一解, ① 所以△=0,即4-4(-k-1)=0.
The root-increasing problems of fractional equations and irrational equations are hot and difficult issues in recent years in the entrance exams and competition propositions. Because such questions do not directly inform candidates of all conditions, they imply that certain conditions are implicit in the problem. In the conclusions or mathematical expressions, when answering, not to mention candidates, that is, mathematics educators can hardly guard against it. Now for example, for reference. Example 1 Equation 2x/(x+1)-k/(x2+x) =x+1/x only has a unique solution, find k. The students answered this way: To denominator, get 2x2-k=(x+1)2, x2-2x-k-1=0. Because the equation has only unique solution , 1 So △=0, that is, 4-4(-k-1)=0.