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初中《代数》第三册P119,习题七第6题:利用根与系数的关系,求一个一元二次方程,使它的根分别是方程2x~2十3x-1=0的各根的①相反数;②倒数;③k倍(为本文之需要,将原题中的“平方”改为“k倍”)。这类题目,同学们一定做了很多。但做完后可曾观察过所求方程和原方程在系数上有何联系,即有无规律可循。下面让我们一起来剖析探讨这个问题: ①、相反数解题按常规解法所求得的新方程为: 2x~2-3x-1==0 (解法略) 现察比较2x~2+3x-1=0和2x~2-3x-1=0。猜想若所求作的方程的两根分别是原方程两根的相反数时,则所求作的方程就是把原方程的一次项系数变号后所得的方程。证明设原方程为ax~2+bx+c=0(a≠0),且两
Junior Middle School Algebra Volume III P119, Problem Set 7 Problem 6: Using the relationship between roots and coefficients, find a quadratic equation with a root that is 1 for each root of equation 2x~2 3 3x-1=0. Inverse number; 2 reciprocal; 3k times (for the purpose of this article, change the “square” in the original question to “k times”). On this type of topic, the students must do a lot. However, after doing so, we have observed whether the equations we sought and the original equations have any relationship with the coefficients, that is, whether there is any law to follow. Let us analyze and discuss this problem together: 1. The new equation obtained by solving the conventional solution according to the inverse number solution is: 2x~2-3x-1==0 (the method is slightly omitted). Compare 2x~2+3x- 1=0 and 2x~2-3x-1=0. It is conjectured that if the two equations of the desired equation are the opposite of the two original equations, the resulting equation is the equation obtained by translating the primary term coefficients of the original equation. Prove that the original equation is ax~2+bx+c=0 (a≠0), and two