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本文試图对一个习題多种解法的問題,作比較系統的分析,供研究这一問题的老师們参考。一、一个习題为什么存在多种解法? 解答数学习題的方法是根据題里的式子或几何图形間的联系而得出的。由于有些习題里的式子或几何图形間的联系是多种多样的,这就决定了一个习題可以有多种解法。例如用图象解方程 x~2-2x-2=0.在这个习題里,方程x~2-2x-2=0与函数y=x~2-2x-2有联系,与函数y=x~2,y=2x-2也有联系。根据这两种联系,就可以得出两种不同的解法: (1)方程x~2-2x-2=0的根是使y=x~2-2x-2函数值为零的自变量x的值。根据这种联系,可得如图1的解法。 (2)方程x~2-2x-2=0的根,是使两个函数y_1=x~2和y_1=2x+2的值相等时自变量x的值。根据这个联系,可得如图2的解法。
This article attempts to make a comparative analysis of the problems of multiple solutions to a problem, and is used as a reference for teachers who study this issue. A. Why do multiple solutions exist for an exercise? The way to solve a number of learning problems is based on the relationship between the equations or the geometry. Because there are many kinds of connections between formulas or geometric shapes in some exercises, this determines that there are many solutions to a problem. For example, use the image to solve the equation x~2-2x-2=0. In this problem, the equation x~2-2x-2=0 is related to the function y=x~2-2x-2, and the function y= x~2, y=2x-2 are also linked. Based on these two connections, two different solutions can be derived: (1) The root of the equation x~2-2x-2=0 is the independent variable x that makes the y=x~2-2x-2 function zero The value of According to this connection, the solution shown in Figure 1 can be obtained. (2) The root of the equation x~2-2x-2=0 is the value of the independent variable x when the two functions y_1=x~2 and y_1=2x+2 are equal. According to this connection, the solution shown in Figure 2 can be obtained.