Factorise $a^{2} - b^{2} - c^{2} + 2bc$.

Factorise $a^{2} - b^{2} - c^{2} + 2bc$.
  1. $(a + b - c)(a - b + c)$
  2. $(a + b - c)(a + b + c)$
  3. $(a - b - c)(a + b + c)$
  4. $(a - b + c)^{2}$

Solution

Group: $a^{2} - (b^{2} - 2bc + c^{2}) = a^{2} - (b-c)^{2} = (a - (b-c))(a + (b-c)) = (a-b+c)(a+b-c)$.

Asked in: IMO

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