如果a=2003x+2001,b=2003x+2002,c=2003x+2003,那么代数式a2+b2+c2-ab-ac-bc的值等于_.
问题描述:
如果a=2003x+2001,b=2003x+2002,c=2003x+2003,那么代数式a2+b2+c2-ab-ac-bc的值等于______.
答
∵a=2003x+2001,b=2003x+2002,c=2003x+2003
∴a-b=-1,b-c=-1,a-c=-2
∴a2+b2+c2-ab-ac-bc
=
(2a2+2b2+2c2−2ab−2ac−2bc )1 2
=
[(a2−2ab+b2)+(b2−2bc+c2)+(a2−2ac+c2)]1 2
=
[(a−b)2+(b−c)2+(a−c)2]1 2
=
(1+1+4)1 2
=3
故答案为3