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解无理方程的常用方法是使方程有理化,但对于一些特殊的无理方程,如果盲目乘方,往往会招致繁琐的运算。这就需要根据题中的一些特殊条件,采用特殊的解法。而利用二次曲线的定义,将无理方程转化为二次曲线的标准方程是值得注意的解题方法,现举几例介绍如下: 例1 解方程 x~2-10(3~(1/2))x+80+(1/2)x~2+10(3~(1/2))x+80=20 解:原方程可化为:(x-5(3~(1/2))~2+5~(1/2)+(x+5(3~(1/2)))~2+5~(1/2)=20令y~2=5,则原方程为:(x-5(3~(1/2))~2+y~2)~(1/2)+(x+5(3~(1/2))~2+y~2~(1/2)=20。此方程表示动点P(x,y)到两定点(5(3~(1/2)),0)、(-5(3~(1/2)),0)的距离之和为20,故它表示椭圆。
The common method for solving irrational equations is to rationalize the equations, but for some special irrational equations, if blind powers are used, tedious operations are often incurred. This requires a special solution based on some special conditions in the question. Using the definition of the quadratic curve, the standard equation that converts the irrational equation into a quadratic curve is a noteworthy problem solving method. Here are a few examples that are introduced as follows: Example 1 Solution equation x~2-10 (3~(1/2) ))x+80+(1/2)x~2+10(3~(1/2))x+80=20 Solution: The original equation can be converted into: (x-5(3~(1/2)) )~2+5~(1/2)+(x+5(3~(1/2)))~2+5~(1/2)=20 Let y~2=5, then the original equation is: (x-5(3~(1/2))~2+y~2)~(1/2)+(x+5(3~(1/2))~2+y~2~(1/ 2) = 20. This equation represents the dynamic point P (x, y) to two fixed points (5 (3 ~ (1/2)), 0), (-5 (3 ~ (1/2)), 0) The sum of distances is 20, so it represents an ellipse.