Express a\(^{2}\) + b\(^{2}\) + c\(^{2}\) - ab - bc - ca as Sum of Squares

Here we will express a\(^{2}\) + b\(^{2}\) + c\(^{2}\) – ab – bc – ca as sum of squares.

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\(^{2}\) + b\(^{2}\) – 2ab) + (b\(^{2}\) + c\(^{2}\) – 2bc) + (c\(^{2}\) + a\(^{2}\) – 2ca)}

                     = \(\frac{1}{2}\){(a - b)\(^{2}\) + (b - c)\(^{2}\) + (c – a)\(^{2}\)}





Corollaries:

(i) If a, b, c are real numbers then (a – b)\(^{2}\), (b – c)\(^{2}\) and (c – a)\(^{2}\) are positive as square of every real number is positive. So,

a\(^{2}\) + b\(^{2}\) + c\(^{2}\) – ab – bc – ca is always positive.

(ii) a\(^{2}\) + b\(^{2}\) + c\(^{2}\) – ab – bc – ca = 0 if \(\frac{1}{2}\){(a - b)\(^{2}\) + (b - c)\(^{2}\) + (c – a)\(^{2}\)} = 0

                                    Or, (a - b)\(^{2}\) = 0, (b - c)\(^{2}\) = 0, (c – a)\(^{2}\)= 0

                                    Or, a - b = 0, b - c = 0, c – a = 0, i.e., a = b = c






9th Grade Math

From Express a^2 + b^2 + c^2 – ab – bc – ca as Sum of Squares to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.