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已知一元二次方程ax2+bx+c=0(a≠0)的两个根为实数,不解方程,求这两个根组成的代数式的值.这是根与系数的一种极为重要的应用,但课本中出现的代数式都是关于两根x1、x2的对称式.所谓关于x1、x2的对称式,是指在代数式中,将x1换成x2,x2换成x1,代数式的值不变.这样的代数式称为关于x1、x2的对称式,如x1x22+x2x12,x13+x23,(x1-x2)2等.如果要求值的代数式不是关于x1、x2的对称式,如x12-3x2,x23+4x12等,如何求它的值?这里介绍一种配偶法.
Knowing that the two roots of the one-dimensional quadratic equation ax2 + bx + c = 0 (a ≠ 0) are real numbers, do not solve the equation and find the algebraic formula of the two roots. This is an extremely important function of the root and the coefficient. The application of algebraic expressions in textbooks is all about the symmetry of x1 and x2. The so-called symmetry about x1 and x2 means that in algebraic expressions, x1 is replaced by x2, x2 is replaced by x1, and algebraic values are used. No change. Such an algebraic expression is called a symmetric expression about x1, x2, such as x1x22+x2x12, x13+x23, (x1-x2)2, etc. If the algebraic expression of the required value is not about the symmetry of x1 and x2, such as x12- 3x2, x23+4x12, etc., how to find its value? Here is a spouse method.