Question

Prove that for any value of a,b,c ; a^2+b^2+c^2-ab-bc-ca is non negative​

Answer 1

Answer:

To prove this equation non-negative, you will have to convert the equation in terms of perfect square form containing a,b and c.

Now,

a²+b²+c²-ab-bc-ca

= ½ • ( 2a²+2b²+2c²-2ab-2bc -2ca )

= ½ • ( a² -2ab +b² +b² -2bc +c² +c² -2ac +a² )  

= ½ • { (a-b)² + (b-c)² + (c-a)² }

For any value of a,b,c  

(a-b)² ≥ 0,

(b-c)² ≥ 0,

(c-a)² ≥ 0,

So,  

a²+b²+c²-ab-bc-ca = ½ • { (a-b)² + (b-c)² + (c-a)² } ≥ 0 i.e. Non-negative [proved]