Question

prove that a square + b square + c square - ab -bc-ca is always neagtive for all values of a b and c.​

Answer 1

Step-by-step explanation:

To prove:-a²+b²+c²-ab-bc-ca is always

positive for all the value of a, b, c.

a²+b²+c²-ab-bc-ca

1/2(2a²+2b²+2c²-2ab-2bc-2ca)

1/2(a²-2ab+b²+b²-2bc+c²+c²-2ac+a²)

1/2((a-b)²+(b-c)²+(c-a)²)

(a-b)² is always positive for all real

values of a,b

(b-c)² is always positive for all real

values of b,c

(c-a)² is always positive for all real

values of c,a

1/2[(a-b)²+(b-c)²+(c-a)²] is always positive for all real values of a, b, c.

Hence proved

Note:-The square of any real number is always positive.