If $a {(3 x - 4 y)}^{2} + 2 b (4 y - 3 x) = (m x - 4 y) (a (m x - 4 y) - 2 b)$ .Find $m$

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Solution

The given expression $a (3 x - 4 y)^{2} + 2 b (4 y - 3 x) = (m x - 4 y) (a (m x - 4 y) - 2 b)$ can be solved as shown below:

$a (3 x - 4 y)^{2} + 2 b (4 y - 3 x) = (m x - 4 y) (a (m x - 4 y) - 2 b) \Rightarrow a (3 x - 4 y)^{2} + 2 b [- (3 x - 4 y)] = (m x - 4 y) (a (m x - 4 y) - 2 b) \Rightarrow a (3 x - 4 y)^{2} - 2 b (3 x - 4 y) = (m x - 4 y) (a (m x - 4 y) - 2 b) \Rightarrow (3 x - 4 y) (a (3 x - 4 y) - 2 b) = (m x - 4 y) (a (m x - 4 y) - 2 b)$

By comparing, we get:

$3 x - 4 y = m x - 4 y \Rightarrow 3 x = m x \Rightarrow m = 3$

Hence, $m = 3$ .

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