Factorise:

$2 (a b + c d) - a^{2} - b^{2} + c^{2} + d^{2}$

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Solution

$2 (a b + c d) - a^{2} - b^{2} + c^{2} + d^{2} \Rightarrow 2 a b + 2 c d - a^{2} - b^{2} + c^{2} + d^{2} \Rightarrow (c^{2} + d^{2} + 2 c d) - (a^{2} + b^{2} - 2 a b) \Rightarrow (c + d)^{2} - (a - b)^{2} \Rightarrow (c + d + a - b) (c + d - a + b)$

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a^{2} + b^{2} - c^{2} - d^{2} + 2 a b - 2 c d

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a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) :

If a, b, c, d are in G.p., prove that :

(i) $(a^{2} + b^{2}), (b^{2} + c^{2}), (c^{2} + d)^{2}$ are in G.P.

(ii) $(a^{2} - b^{2}), (b^{2} - c^{2}), (c^{2} - d)^{2}$ are in G.P.

(iii) $\frac{1}{a^{2} + b^{2}}, \frac{1}{b^{2} + c^{2}}, \frac{1}{c^{2} + d^{2}}$ are in G.P.

(iv) $(a^{2} + b^{2} + c^{2}), (a b + b c + c d), (b^{2} + c^{2} + d^{2})$

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