Question

已知a-b=√3-√2，b-c=√3+√2，求2（a^2+b^2+c^2-ab-bc-ac)

Answer 1

a-b=√3-√2，b-c=√3+√2
l两式相加，得a-b+b-c=2√3，即a-c=2√3
2（a^2+b^2+c^2-ab-bc-ac)
=2a²+2b²+2c²-2ab-2bc-2ac
=a²-2ab+b²+b²-2bc+c²+a²-2ac+c²
=(a-b)²+(b-c)²+(a-c)²
=(√3-√2)²+(√3+√2)²+(2√3)²
=3-2√6+2+3+2√6+2+12
=22

Answer 2

2（a^2+b^2+c^2-ab-bc-ac)
=a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ac+a^2
=(a-b)^2+(b-c)^2+(c-a)^2
=(√3-√2)^2+(√3+√2)^2+(2√3)^2
=10+12
=22

Answer 3

a-b=√3-√2，b-c=√3+√2
得a-b+b-c=2√3，即a-c=2√3
2（a^2+b^2+c^2-ab-bc-ac)
=2a²+2b²+2c²-2ab-2bc-2ac
=a²-2ab+b²+b²-2bc+c²+a²-2ac+c²
=(a-b)²+(b-c)²+(a-c)²
=(√3-√2)²+(√3+√2)²+(2√3)²
=3-2√6+2+3+2√6+2+12
=22

Answer 4

A-B=根号3+根号2，B-C=根号3-根号2,
A-C=2根号3
A^2+B^2+C^2-AB-BC-AC
=[(A-B)^2+(B-C)^2+(A-C)^2]/2
=(5+2*6^(1/2)+5-2*6^(1/2)+12)/2=11