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If    a2+b2+4c2 = ab+2bc+2ca | Expansion - Algebra

If    a2+b2+4c2 = ab+2bc+2ca.    Prove that a = b = 2c

Multiplying both side by 2 we have.

2. a2 + 2. b2 + 2.4c2 = 2.ab + 2.2bc + 2.2ca

2. a2 + 2. b2 + 2.4c2 = 2ab + 4bc + 4ca
2. a2 + 2. b2 + 2.4c2 - 2ab - 4bc - 4ca =0
( a2 - 2ab + b2 ) + ( b2 - 4bc + 4c2 ) + ( a2 - 4ca + 4c2 ) = 0
( a - b )2 + (b - c)2 + (a- 2c)2 = 0
If the Sum of the squares will be equal to zero then the individual terms will also be equal to zero.
∴ ( a - b )2 = 0
(b - c)2 = 0
and (a- 2c)2 = 0
Now ( a - b )2 = 0 , or ( a - b ) = 0 or a = b
Similarly (b - c)2 = 0 , or (b - c) = 0 or b = c
Similarly (a- 2c)2 = 0 , or (a- 2c) = 0 . or a = 2c
∴ a = b = 2c Proved

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