Question :

If the roots of the equation ( a − b ) ² + ( b − c ) ² + ( c − a )² = 0 are equal, prove that 2 a = b + c .Submitted on 8/3/2024 | Answered by Vandana Rana

Answer :

Given, ( a − b ) x2 + ( b − c ) x + ( c − a ) = 0
are equal.
Then the discriminant (D) = 0
Then ( b − c )2 − 4 ( c − a ) ( a − b ) = 0
b2 − 2 b c + c2 − 4 ( a c − b c − a2 + a b ) = 0
b2 - 2 b c + c2 − 4ac + 4bc + 4a2 - 4 ab=0
b2 + c2 + 4a2 + 2bc - 4ac - 4ab =0
( b + c − 2 a )2 = 0
2 a = b + c