The perimeter of a rectangle is $8 a^{2} - 10 a b + 6 b^{2}$ and one of its side is $6 a^{2} - 15 a b + 2 b^{2}$ . Find its other adjacent side.

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Solution

$P e r i m e t e r = 2 (x + y)$
where,
$x =$ one side
$y =$ other side
So,
$8 a^{2} - 10 a b + 6 b^{2} = 2 (6 a^{2} - 15 a b + 2 b^{2} + y)$
$8 a^{2} - 10 a b + 6 b^{2} = 12 a^{2} - 30 a b + 4 b^{2} + 2 y$
$8 a^{2} - 10 a b + 6 b^{2} - 12 a^{2} + 30 a b - 4 b^{2} = 2 y$
$2 y = - 4 a^{2} + 20 a b + 2 b^{2}$
$y = \frac{- 4 a^{2} + 20 a b + 2 b^{2}}{2}$
$y = - 2 a^{2} + 10 a b + b^{2}$

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