Factorise:
$9 x^{2} + 6 x y + y^{2}$

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Solution

Given $9 x^{2} + 6 x y + y^{2}$
Express the middle term $6 x y$ as sum of two terms such that their product is equals the product of extreme terms $(9 x^{2} y^{2})$ .
So, $6 x y = 3 x y + 3 x y$ and also $3 x y \times 3 x y = 9 x^{2} y^{2}$
Thus, the given expression can be written as
$= 9 x^{2} + 3 x y + 3 x y + y^{2}$
$= 3 x (3 x + y) + y (3 x + y)$
$= (3 x + y) (3 x + y)$
$= (3 x + y)^{2}$
Therefore, the factorised form is $(3 x + y)^{2}$ .

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