Prove that
$(2.5 p - 1.5 q)^{2} - (1.5 p - 2.5 q)^{2}$

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Solution

$(2.5 p - 1.5 q)^{2} - (1.5 p - 2.5 q)^{2}$

$= (2.5 p)^{2} - 2 (2.5 p) (1.5 q) + (1.5 q)^{2} - [(1.5 p)^{2} - 2 (1.5 p) (2.5 q) + (2.5 q)^{2}] [(a - b)^{2} = a^{2} - 2 a b + b^{2}]$

$= 6.25 p^{2} - 7.5 p q + 2.25 q^{2} - [2.25 p^{2} - 7.5 p q + 6.25 q^{2}]$

$= 6.25 p^{2} - 7.5 p q + 2.25 q^{2} - 2.25 p^{2} + 7.5 p q - 6.25 q^{2}]$

$= 4 p^{2} - 4 q^{2}$

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