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Factorisation Using Algebraic Identities

Factorise: ...

Question

Factorise: $36 {(5 x + y)}^{2} - 25 {(4 x - y)}^{2}$

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Solution

We know that $a^{2} - b^{2} = (a + b) (a - b)$

Using the above identity, the Expression $36 (5 x + y)^{2} - 25 (4 x - y)^{2}$ can be factorised as follows:

$36 (5 x + y)^{2} - 25 (4 x - y)^{2} = (6 (5 x + y))^{2} - (5 (4 x - y))^{2} = [6 (5 x + y) - 5 (4 x - y)] [6 (5 x + y) + 5 (4 x - y)]$
$= (30 x + 6 y - 20 x + 5 y) (30 x + 6 y + 20 x - 5 y) = (10 x + 11 y) (50 x + y)$

Hence, $36 (5 x + y)^{2} - 25 (4 x - y)^{2} = (10 x + 11 y) (50 x + y)$

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