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数学的定义是建立数学大厦的基石,求与一元二次方程的根有关的代数式之值的问题时,若能恰当地用根的定义来解,则简捷明快,事半功倍.一、求代数式的值例1若m、n是关于x的方程x~2+(p一2)x+1=0的两个根,求代数式(m~2+mp+1)(n+np+1)的值.析解若展开变形求解,则相当繁冗.但依题意易想到方程根的定义,有m~2+(p-2)m+1=0,n~2+(p-2)n+1=0.再观察待求式,又可想到将此二式继而变形为m~2+mp+1=2m,
The definition of mathematics is the cornerstone of building a mathematics building. When you solve the problem of the algebraic formula related to the root of a one-dimensional quadratic equation, if you can properly solve it with the definition of the root, it is simple and quick, and you can do more with less. 1. Find algebraic expressions Example 1 If m and n are two roots of the equation x~2+(p−2)x+1=0 for x, find the value of the algebraic expression (m~2+mp+1)(n+np+1). Analysis of the solution to the expansion of the deformation, it is quite tedious. However, according to the meaning of the idea of the root of the equation is easy to think, there are m~2+(p-2)m+1=0,n~2+(p-2)n+ 1 = 0. After observing the formula, it is also conceivable that this formula is then transformed into m~2+mp+1=2m.