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对型如((x-a)~2+b)~(1/2)+((c-x)~2+d)~(1/2)=k的无理方程,可构造直角三角形,运用勾股定理和相似形,使之转化为简单的方程组来解,堪为巧妙! 例1 解方程 (x~2+1)~(1/2)+(x~2-24x+160)~(1/2)=13。解原方程可化为: (x~2+1)~(1/2)+((12-x)~2+16)~(1/2)=13。令y=12-x,则有(x~2+1)~(1/2)+(y~2+16)~(1/2)=13 如图1,构造直角△ABC,使∠C=90°,AC=12,AB=13,则BC=(13~2-12~2)~(1/2)=5
For irrational equations of type ((xa)~2+b)~(1/2)+((cx)~2+d)~(1/2)=k, a right-angled triangle can be constructed using the Pythagorean theorem and It is clever to make a similar equation into a simple equation group. Example 1 Solution equation (x~2+1)~(1/2)+(x~2-24x+160)~(1/2) )=13. The original equation can be solved as: (x~2+1)~(1/2)+((12-x)~2+16)~(1/2)=13. Let y=12-x, then there is (x~2+1)~(1/2)+(y~2+16)~(1/2)=13. As shown in Fig. 1, the right angle △ABC is constructed so that ∠C =90°, AC=12, AB=13, then BC=(13~2-12~2)~(1/2)=5