Question

Prove thatt a^2+b^2+c^2-ab-bc-ca is always non negative for all values of a,b,c

Answer 1

the power is in even
so every negative or positive integer result in positive

Answer 2

a² + b² + c² - ab - bc - ca

Taking ½ common,

½ (2a² + 2b² + 2c² - 2ab - 2bc - 2ca)

= ½ {(a² + b² - 2ab) + (b² + c² - 2bc) + (c² + a² - 2ca)}

= ½ {(a - b)² + (b - c)² + (c - a)²}

The sum of perfect squares is always non-negative.

Since the half of non-negative number is always non-negative,

a² + b² + c² - ab - bc - ca is always non-negative for all values of a, b and c.

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