Factorize the following algebraic expression:

$25 - p^{2} - q^{2} - 2 p q$

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Solution

Given: $25 - p^{2} - q^{2} - 2 p q$

$= 25 - (p^{2} + q^{2} + 2 p q)$

Using, $(a + b)^{2} = a^{2} + b^{2} + 2 a b$

$= 25 - (p + q)^{2}$

$= (5)^{2} - (p + q)^{2}$

Using, $a^{2} - b^{2} = (a + b) (a - b)$

$= [5 - (p + q)] [5 + (p + q)]$

$= (5 - p - q) (5 + p + q)$

$= - (p + q - 5) (p + q + 5)$

$∴ 25 - p^{2} - q^{2} - 2 p q = - (p + q - 5) (p + q + 5)$

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