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例1(文[1]例3)设a>b>c且a+b+c=1,a~2+b~2+c~2=1,求a+b的取值范围.文[1]构造了以a,b为根的一元二次方程,利用方程有两个不等实根的条件,去求c的取值范围,进而求出a+b的取值范围,解法比较自然,易于想到,但因为没考虑条件a>b>c,出现了错误.文[2]指出了文[1]的错误以及出错的原因,并且给出了新的构造方程的方法,但解法稍显繁琐,且不太容易想到.本文将对文[1]的方法进行补充和完善,使其简便易行.
Example 1 (Example 1 in [1]) finds the range of a + b for a> b> c and a + b + c = 1 and a ~ 2 + b ~ 2 + c ~ 2 = 1] constructs a quadratic equation a, b roots, the equation has two unequal real conditions, to find the range of c, and then find the range of a + b, the solution is more natural , But it is easy to think about it, but because we did not consider the condition a> b> c, we got an error. Man [2] pointed out the error of the paper [1] and the reason of the error, and gave a new method to construct the equation. Is cumbersome, and not easy to think of.This article will complement and improve the method [1], making it easy and convenient.