Factorise the following using appropriate identities:
(i) $9 x^{2} + 6 x y + y^{2}$
(ii) $4 y^{2} - 4 y + 1$
(iii) $x^{2} - \frac{y^{2}}{100}$

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Solution

(i) $9 x^{2} + 6 x y + y^{2} = (3 x)^{2} + 2 (3 x) (y) + (y)^{2} = (3 x + y)^{2}$
(ii) $4 y^{2} - 4 y + 1 = (2 y)^{2} - 2 (2 y) (1) + 1 = (2 y - 1)^{2}$
(iii) $x^{2} - \frac{y^{2}}{100} = (x)^{2} - {(\frac{y}{10})}^{2} = (x - \frac{y}{10}) (x + \frac{y}{10})$

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