Factorize the following algebraic expression:

$x^{2} - y^{2} + 6 y - 9$

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Solution

Given: $x^{2} - y^{2} + 6 y - 9$
$= x^{2} - (y^{2} - 6 y + 9)$
$= x^{2} - [y^{2} - 2 (y) (3) + (3)^{2}]$
Using, $(a - b)^{2} = a^{2} - 2 a b + b^{2}$
$= x^{2} - (y - 3)^{2}$
Using, $a^{2} - b^{2} = (a + b) (a - b)$
$= (x + y - 3) [x - (y - 3)]$
$= (x + y - 3) (x - y + 3)$

$∴ x^{2} - y^{2} + 6 y - 9 = (x + y - 3) (x - y + 3) .$

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