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解分式方程,可能产生增根,这是大家熟知的事。然而,用下面的方法解分式方程,竞出现失根。例解方程((18+x)~(1/2)+(x-5)~(1/2))/((18+x)~(1/2)+(x-5)~(1/2))= =((10-x)~(1/2)+((x-5)~(1/2))/((10-x)~(1/2)+((x-5)~(1/2)) 解用合分比定理化方程为 ((18+x)~(1/2))/(x-5)~(1/2))=((+x)/(1/2)/(x-5)~(1/2)) 两边平方,整理得 2x=-8,x=-4。经检验,-4是原方程的增根。是不是原方程无根呢?不是的。原方程还有x=5这一根被遗失了。可见用合分比定理解分式方程可能失根。以下研究失根的原因。
It is well known that solving fractional equations may produce rooting. However, using the following method to solve the fractional equation, you lose ground. The solution equation ((18+x)~(1/2)+(x-5)~(1/2))/((18+x)~(1/2)+(x-5)~(1 /2)) = =((10-x)~(1/2)+((x-5)~(1/2))/((10-x)~(1/2)+((x- 5)~(1/2)) Solve the equation of the theorem with the fractional ratio ((18+x)~(1/2))/(x-5)~(1/2))=((+x) /(1/2)/(x-5)~(1/2)) Squared on both sides, collated to 2x=-8, x=-4. Upon examination, -4 is the root of the original equation. Is it the original equation? No roots? No. The original equation and x=5 root are missing. It can be seen that using fractional ratios to understand fractional equations may lead to root loss.