Toán Lớp 8: Cho a^3+b^3+c^3=3abc
CMR: a+b+c=0 hoặc a=b=c
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Toán Lớp 8: Cho a^3+b^3+c^3=3abc
CMR: a+b+c=0 hoặc a=b=c
Comments ( 2 )
<=>a^3+b^3+c^3-3abc=0
<=>a^3+3a^2b+3ab^2+b^3+c^3-3a^2b-3ab^2-3abc=0
<=>(a+b)^3+c^3-3ab(a+b+c)=0
<=>(a+b+c)[(a+b)^2-(a+b)c+c^2]-3ab(a+b+c)=0
<=>(a+b+c)(a^2+2ab+b^2-ca-bc+c^2-3ab)=0
<=>(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0
+)Th1 :
a^2+b^2+c^2-ab-bc-ca=0
<=>2(a^2+b^2+c^2-ab-bc-ca)=0
<=>2a^2+2b^2+2c^2-2ab-2bc-2ca=0
<=>(a^2-2ab+b^2)+(b^2-2bc+c^2)+(c^2-2ca+a^2)=0
<=>(a-b)^2+(b-c)^2+(c-a)^2=0
<=>$\begin{cases}(a-b)^2=0\\(b-c)^2=0\\(c-a)^2=0\end{cases}$
<=>$\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}$
+)Th2 :
a+b+c=0