Question
Factorize:
$x^2 - y^2 - 4xz + 4z^2$
$x^2 - y^2 - 4xz + 4z^2$
✓
Answer
$x^2-y^2-4 x z+4 z^2$
On rearranging the terms $=x^2-4 x z+4 z^2-y^2=(x)^2-2 \times x \times 2 z+(2 z)^2-y^2$
Using the identity $x^2-2 x y+y^2=(x-y)^2=(x-2 z)^2-y^2$
Using the identity $p^2-q^2=(p+q)(p-q)$
$=(x-2 z+y)(x-2 z-y)$
$\therefore x^2-y^2-4 x z+4 z^2=(x-2 z+y)(x-2 z-y)$
On rearranging the terms $=x^2-4 x z+4 z^2-y^2=(x)^2-2 \times x \times 2 z+(2 z)^2-y^2$
Using the identity $x^2-2 x y+y^2=(x-y)^2=(x-2 z)^2-y^2$
Using the identity $p^2-q^2=(p+q)(p-q)$
$=(x-2 z+y)(x-2 z-y)$
$\therefore x^2-y^2-4 x z+4 z^2=(x-2 z+y)(x-2 z-y)$
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