Simplify the following:
$(7 a / 2 - 5 b / 2)^{2} - 5 a / 2 - 7 b / 2)^{2}$

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Solution

$(7 a / 2 - 5 b / 2)^{2} - 5 a / 2 - 7 b / 2)^{2}$
$[(\frac{7}{2} a)^{2} - 2 \times \frac{7}{2} a \times \frac{5}{2} b - (\frac{5}{2} b)^{2}] - [(\frac{5}{2} a)^{2} - 2 \times \frac{5}{2} a \times \frac{7}{2} b + (\frac{7}{2} b)^{2}]$
$= [\frac{49}{4} a^{2} - \frac{35}{2} a b + \frac{25}{4} b^{2}] - [\frac{25}{4} a^{2} - \frac{35}{2} a b + \frac{49}{4} b^{2}]$
$= \frac{49}{4} a^{2} - \frac{25}{4} a^{2} + \frac{25}{4} b^{2} - \frac{49}{4} a^{2}$
$= \frac{24}{4} a^{2} + \frac{- 24}{4} b^{2}$
$= 6 a^{2} - 6 b^{2}$

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