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 $\mathrm{a}^2-\mathrm{b}^2=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})$
$=(a+b-c)(a-(b-c))$
$=(a+b-c)(a-b+c)$
$\therefore a^2-b^2+2 b c-c^2$
$=(a+b-c)(a-b+c)$
$a^2-\left(b^2-2 b c+c^2\right)$
Using the identity $(a-b)^2=a^2+b^2-2 a b$
$=a^2-(b-c)^2$
Using the identity $\mathrm{a}^2-\mathrm{b}^2=(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})$
$=(a+b-c)(a-(b-c))$
$=(a+b-c)(a-b+c)$
$\therefore a^2-b^2+2 b c-c^2$
$=(a+b-c)(a-b+c)$
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