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解题过程就是不断地将未知转化为已知的过程。而“构造法”则是实现这种转化的重要手段之一。它的策略思想是,对于一个较为复杂(甚至看来无从下手)的问题。先构造一个与之有关的辅助命题,也就是在已知与未知间搭一个桥,借以沟通“条件”和“结论”。例1 解方程 ((3x~2-5x-12)~(1/2))-(2x~2-11x+15)~(1/2)=x-3。解令((3x~2-5x-12)~(1/2))=u;(2x~2-11x+15)~(1/2)=υ;则 u-υ=x-3 ①当u=v时,x=3.代入原方程检验。知x_1=3是它的根。当u≠v时,u+v=(u~2-v~2)/(u-v)=(x~2+6x-27)/(x-3)=x-9②由①和②得u=x+3.υ=6
The problem-solving process is to constantly convert unknowns into known processes. The “construction method” is one of the important means to achieve this transformation. Its strategic thinking is for a more complex (even seemingly impossible) problem. First construct an auxiliary proposition related to it, that is, build a bridge between known and unknown to communicate “conditions” and “conclusions.” Example 1 Solution equation ((3x~2-5x-12)~(1/2))-(2x~2-11x+15)~(1/2)=x-3. The order ((3x~2-5x-12)~(1/2))=u; (2x~2-11x+15)~(1/2)=υ; then u-υ=x-3 1 When u=v, x=3. Substituting into the original equation test. Knowing x_1=3 is its root. When u≠v, u+v=(u~2-v~2)/(uv)=(x~2+6x-27)/(x-3)=x-92 is derived from 1 and 2 u= x+3.υ=6