有趣的数学【19】:色即是空——用待定系数法解方程
人生麻烦不断,就和数学题一样,总有人出那种千奇百怪的题目来考我们。
很多时候,我们一下子找不到答案,只能先假设待定一些前提,然后抽丝剥茧慢慢来,终会有水落石出的那一天。
我们先看这道题,一个式子,有2个未知数,简直就是对人性的摧残:
这道题,用待定系数法来求解,会很简单。
待定系数法和配方法其实差不多,针对一个复杂的多项式,比如上面这种,如果我们可以把它简化成(……)² + (……)² + (……)² = 0的这种形式(平方和),就方便多了。
因为x,y都是实数,所以(……)²是大于等于0的,所以上面的括号内的算式就等于0。
不管来什么麻烦,任何妖魔鬼怪,最后都让它们化为虚无,成为0。色即是空,麻烦就被解决了。
我们现在试着把题目中复杂的多项式变形成平方和的形式:
假设x=a,y=b,
则x-a = 0,
y-b = 0;
观察原题多项式,有xy的乘积项,所以配出的平方项一定含有(x+y)部分。而要(x+y+……)²=0,则应该是:
(x+y - (a+b))² = 0
而(x+y - (a+b))²这一项展开后,有x²和y²项,根据和原题,是5x²和2y²,所以还要有4个x²和1个y²。所以,原式就配成了:
进一步展开,并且合并同类项,则有:
因为都等于0,所以这个式子和原式是相等的,则有:
看到没有,这就是色即是空的魅力,诱惑当前迷人双眼,只有不为所动让其归零,才能让麻烦化为虚无。
附上解题步骤:
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