若a+2b+3c=12,且a2+b2+c2=ab+bc+ca,则a+b2+c3=_.

问题描述:

若a+2b+3c=12,且a2+b2+c2=ab+bc+ca,则a+b2+c3=______.

∵a2+b2+c2=ab+bc+ca,∴2(a2+b2+c2)=2(ab+bc+ca),即2(a2+b2+c2)-2(ab+bc+ca)=0,整理,得(a2-2ab+b2)+(a2-2ca+c2)+(b2-2bc+c2)=0,即:(a-b)2+(a-c)2+(b-c)2=0,∴a=b=c,又∵a+2b+3c=12,∴a...