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Properties of Square Numbers

Prove the fol...

Question

Prove the followings,
i) $(a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})$
ii) $(a + b)^{2} - (a - b)^{2} = 4 a b$

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Solution

We have,
i) $(a + b)^{2} + (a - b)^{2}$
$= (a^{2} + 2 a b + b^{2}) + (a^{2} - 2 a b + b^{2})$
$= 2 (a^{2} + b^{2})$

ii) $(a + b)^{2} - (a - b)^{2}$
$= (a^{2} + 2 a b + b^{2}) - (a^{2} - 2 a b + b^{2})$
$= a^{2} + 2 a b + b^{2} - a^{2} + 2 a b - b^{2}$
$= 4 a b$ .

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