show that if (a - b)^2 = 2(a^2 + b^2) , then a = -b

Question

using algebraic identities

Answer 1

Given (a-b)^2 = 2(a^2+b^2)

We know that, (a-b)^2 = a^2 + b^2 - 2ab

So given equation can be written like this

a^2 + b^2 - 2ab = 2(a^2 + b^2)

a^2 + b^2 - 2ab = 2a^2 + 2b^2

-2ab = a^2 + b^2

a^2 + b^2 + 2ab = 0

(a+b)^2 = 0 ( ∵ (a+b)^2 = a^2 + b^2 + 2ab)

a + b = 0

a = -b

Hence its proved

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