If (a-b)^2 + (b-c)^2 = 0.then prove that a=b=c.

Question

The question is a proving type question of class 8 standard.

Answer 1

Given

 

(a - b)² + (b - c)² = 0

 

(a - b)² = -(b - c)²

 

But (a - b)² ≥ 0 and (b - c)² ≥ 0

 

Then (a - b)² = 0 and (b - c)² = 0

 

Therefore, a = b, and b = c.

Answer 2

Since any squaring of a real expression equals 0 or more (never negative), both (b-c )and (a-b) must each be zero.

 

Therefore b-c = 0

and         a-b = 0

 

The first implies b=c; the second implies a=b; therefore a=b=c by the transitive property of equality.