If $2 a = 3 + 2 b,$ then $8 a^{3} - 8 b^{3} - 36 a b = m$ .Find $m$

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Solution

$8 a^{3} - 8 b^{3} - 36 a b = (2 a)^{3} - (2 b)^{3} - 36 a b = (3 + 2 b)^{3} - (2 b)^{3} - 36 \times (\frac{3 + 2 b}{2}) b = 27 + (2 b)^{3} + 36 b^{2} + 54 b - (2 b)^{3} - 54 b - 36 b^{2} = 27$

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