Factorise $48 a^{2} - 243 b^{2}$

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Solution

We see that the two terms are not perfect squares. But both has '3' as common factor.
That is $48 a^{2} - 243 b^{2} = 3 [16 a^{2} - 81 b^{2}]$
$= 3 [(4 a)^{2} - (9 b)^{2}]$ Again $a^{2} - b^{2} = (a + b) (a - b)$
$= 3 [(4 a + 9 b) (4 a - 9 b)]$
$= 3 (4 a + 9 b) (4 a - 9 b)$

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