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Question

If a+b+c=12 and a2+b2+c2=50, find ab+bc+ca

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Solution

Using the identity (a+b+c)2=a2+b2+c2+2ab+2bc+2ca.

It is given that a+b+c=12 and a2+b2+c2=50, therefore,

(a+b+c)2=a2+b2+c2+2ab+2bc+2ca(a+b+c)2=a2+b2+c2+2(ab+bc+ca)(12)2=50+2(ab+bc+ca)144=50+2(ab+bc+ca)2(ab+bc+ca)=144502(ab+bc+ca)=94ab+bc+ca=942=47

Hence, ab+bc+ca=47.

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