已知a-b=3,b-c=5,求a²+b+c-ab-bc-ca的值

题目

已知a-b=3,b-c=5,求a²+b+c-ab-bc-ca的值

答案

∵a-b=3,b-c=5 两式相加得
∴a-c=8
a²+b²+c²-ab-bc-ca
=1/2(a²-2ab+b²)+1/2(b²-2bc+c²)+1/2(c²-2ca+a²)
=1/2(a-b)²+1/2(b-c)²+1/2(c-a)²
=1/2×3²+1/2×5²+1/2×8²
=1/2×9+1/2×25+1/2×64
=1/2×(9+25+64)
=1/2×98
=49
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