Question
Factorize:
$a^2-b^2+2 b c-c^2$
$a^2-b^2+2 b c-c^2$
✓
Answer
$a^2-b^2+2 b c-c^2 a^2-\left(b^2-2 b c+c^2\right)$
Using the identity $(a-b)^2$
$=a^2+b^2-2 a b$
$=a^2-(b-c)^2$
Using the identity $a^2-b^2$
$=(a+b)(a-b)$
$=(a+b-c)(a-(b-c))$
$=(a+b-c)(a-b+c)$
$\therefore a^2-b^2+2 b c-c^2$
$=(a+b-c)(a-b+c)$
Using the identity $(a-b)^2$
$=a^2+b^2-2 a b$
$=a^2-(b-c)^2$
Using the identity $a^2-b^2$
$=(a+b)(a-b)$
$=(a+b-c)(a-(b-c))$
$=(a+b-c)(a-b+c)$
$\therefore a^2-b^2+2 b c-c^2$
$=(a+b-c)(a-b+c)$
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