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关于二次三项式ax~2+bx+c(a≠0),本文主要研究两个方面的问题: 一、二次三项式能因式分解的判定二次三项式ax~2+bx+c(a≠0)在给定数集内能否进行因式分解,这是中学代数的一个重要课题。现介绍如下四个定理。定理一有理系数二次二项式ax~2+bx+c(a≠0)在有理数集内能分解因式的充要条件是△=b~2-4ac为一个有理效的平方。证明:(1)必要性,若 ax~2+bx+c=a(x-x_1)(x-x_2),为有理数,因a,b为有理数x_1,x_2也为有理数,故只有(b~2-4ac)~(1/2)为有理数。设(b~2-4ac=|m|(m为有理数),则b~2-4ac=m~2。即判别式△=b~2-4ac是一个有理数的平方。
With regard to quadratic trinomials ax~2+bx+c(a≠0), this paper mainly studies two aspects: The first and second trinomial can be factorized to determine the second trinomial ax~2+ Whether or not bx + c (a ≠ 0) can be factorized within a given number set is an important issue for algebra in middle schools. The following four theorems are introduced. Theorem-a rational coefficient The quadratic binomial ax~2+bx+c(a≠0) can be factorized in a rational number set. The necessary and sufficient condition is that △=b~2-4ac is a rational square. Proof: (1) Necessity, if ax~2+bx+c=a(x-x_1)(x-x_2) is a rational number, because a, b is a rational number x_1, and x_2 is also a rational number, so only (b~ 2-4ac)~(1/2) is a rational number. Let (b~2-4ac=|m|(m is a rational number), then b~2-4ac=m~2. That is, the discriminant △=b~2-4ac is the square of a rational number.