Question

设a,b,c为△ABC的三边，求证：关于x的方程x²+2ax+b²＝0与x²-2cx-b²＝0有公共根的充要

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设a,b,c为△ABC的三边，求证：关于x的方程x²+2ax+b²＝0与x²-2cx-b²＝0有公共根的充要条件是角A等于90°

Answer 1

充分性
因为A等于90°
所以b2+c2=a2代入x²+2ax+b²＝0与x²-2cx-b²＝0中得x²+2ax+ a2- c2＝0（x+a）2= c2
得到
x²-2cx-b²＝0得到x²-2cx- a2+ c2＝0得到（x-c）2= a2
得出公共根。
必要性
先解方程x²+2ax+b²＝0与x²-2cx-b²＝0得到x=(-b2)/(a+c)，代入第一个方程求出勾股定理的式子

Answer 2

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Answer 3

必要性：第一个方程减去第二个方程解得x=(-b^2)/(a+c)，代入第二个方程，整理化简立得b^2+c^2=a^2。
充分性：显然。