Which of the following is a factorisation of $9 x^{2} + 2 x y + \frac{y^{2}}{9}$ ?

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Solution

$9 x^{2} + 2 x y + \frac{y^{2}}{9}$
$= (3 x)^{2} + 2 (3 x) (\frac{y}{3}) + {(\frac{y}{3})}^{2}$
$= {(3 x + \frac{y}{3})}^{2} [∵ a^{2} + 2 a b + b^{2} = (a + b)^{2}$
$= (3 x + \frac{y}{3}) (3 x + \frac{y}{3})$

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