Question

if a^2 + b^2 + c^2 -ab -bc -ca = 0, Show that a=b=c.

Answer 1

It is given that a2+b2+c2=0.

I am assuming that a,b and c are all real numbers.

The square of a real number cannot be negative.

The sum of non-negative numbers cannot be equal to zero unless all the numbers are individually equal to zero.

Therefore, a2+b2+c2=0⇒a=0,b=0 and c=0.

⇒a3=0,b3=0 and c3=0.

⇒a3+b3+c3=0 and abc=0.

⇒a3+b3+c3=3abc.

Answer 2

a^2+b^2+c^2-ab-bc-ca=0

(a+b+c)^2=0

a+b+c=0

here, sum of a,b,c is 0

so,a=b=c.