Factorise: $25 a^{2} - 4 b^{2} + 28 b c - 49 c^{2}$

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Solution

$25 a^{2} - 4 b^{2} + 28 b c - 49 c^{2}$
$= (5 a)^{2} - (4 b^{2} - 28 b c + 49 c^{2})$
$= (5 a)^{2} - [(2 b)^{2} - 2 (2 b) (7 c) + (7 c)^{2}$ ] [Using $x^{2} - 2 x y + y^{2} = (x - y)^{2}$ ]
$= (5 a)^{2} - (2 b - 7 c)^{2}$
$= (5 a - 2 b + 7 c) (5 a + 2 b - 7 c)$ [Using $x^{2} - y^{2} = (x - y) (x + y)$ ]
Hence, $25 a^{2} - 4 b^{2} + 28 b c - 49 c^{2} = (5 a - 2 b + 7 c) (5 a + 2 b - 7 c)$

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