Question

ifa=2012,b=2011,c=2010 find a^2+b^2+c^2-ab-bc-ca
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Answer 1

Answer:

$\underline{Given}:a=2012,\: b=2011\: and\: c=2010\\ \\ \underline{To\: Find}:a^{2}+b^{2}+c^{2}-ab-bc-ca\\ \\ \underline{Solution}:a^{2}+b^{2}+c^{2}-ab-bc-ca\\ =\frac{1}{2}[2(a^{2}+b^{2}+c^{2}-ab-bc-ca)]\\ =\frac{1}{2}(2a^{2}+2b^{2}+2c^{2}-2ab-2bc-2ca)\\ =\frac{1}{2}(a^{2}+a^{2}+b^{2}+b^{2}+c^{2}-2ab-2bc-2ca)\\ =\frac{1}{2}(a^{2}-2ab+b^{2}+b^{2}-2bc+c^{2}+c^{2}-2ca+a^{2})\\ =\frac{1}{2} [(a-b)^{2}+(b-c)^{2}+(c-a)^{2}]\\ =\frac{1}{2} [(2012-2011)^{2}+(2011-2010)^{2}+(2010-2012)^{2}]\\ =\frac{1}{2} (1^{2}+1^{2}+(-2)^{2})\\ =\frac{1}{2} (1+1+4)\\ =\frac{1}{2}(6)\\ =3_____\bold{(ans)}$