已知实数a,b,c满足abc=-1,a+b+c=4,aa
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已知实数a,b,c满足abc=-1,a+b+c=4,
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∵abc=-1,a+b+c=4,
∴a2-3a-1=a2-3a+abc=a(bc+a-3)=a(bc-b-c+1)=a(b-1)(c-1), ∴ a a2−3a−1= 1 (b−1)(c−1), 同理可得: b b2−3b−1 = 1 (a−1)(c−1), c c2−3c−1= 1 (a−1)(b−1), 又 a a2−3a−1+ b b2−3b−1 + c c2−3c−1= 4 9, ∴ 1 (b−1)(c−1)+ 1 (a−1)(c−1)+ 1 (a−1)(b−1)= 4 9, ∴ (a−1)+(b−1)+(c−1) (a−1)(b−1)(c−1)= 4 9,即 4 9(a-1)(b-1)(c-1)=(a-1)+(b-1)+(c-1), 整理得: 4 9(abc-ab-ac-bc+a+b+c-1)=a+b+c-3, 将abc=-1,a+b+c=4代入得:ab+bc+ac=- 1 4, 则a2+b2+c2=(a+b+c)2-2(ab+bc+ac)= 33 2. 故答案为: 33 2.
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