Negative or positive

Algebra Level 2

If a,b and c are real no.s and a≠b then the value of a 2 a^{2} + b 2 b^{2} + c 2 c^{2} - ab - bc - ca will be always

negative imaginary positve not certain

Manish Mayank by Manish Mayank , 20, India

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2 solutions

Manish Mayank
Jul 14, 2014

You will find that it can also be written as ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 2 \frac{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}{2} . Now squares of reals is always positive, sum of positives is always positive, and half of positive is always positive.

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The answer to this problem is wrong, it should be 'non-negative', since the expression can also reach 0 0 when a = b = c a=b=c . Squares of reals are not always positive (as you claim in this solution), e.g. 0 2 ≯ 0 0^2\not > 0 , so your statement is incorrect. They are indeed always non-negative, though.

mathh mathh - 6 years, 11 months ago

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Good. Happy that you brought up this point :)

Krishna Ar - 6 years, 11 months ago

Yes I observed that

Manish Mayank - 6 years, 11 months ago

Thank you. After your feedback, Manish added the condition that a ≠ b a \neq b .

Calvin Lin Staff - 6 years, 11 months ago
Krishna Garg
Aug 12, 2014

considering real numbers and a is not equal to b,if we take a=1,b=2,c=3,or a=2,b=3,c=4,wer always get the values as +ve.Ans K.K.GARG,India

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