Mathematics
Factorise the following:
(a2 - b2)(c2 - d2) - 4abcd
Factorisation
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Answer
(a2 - b2)(c2 - d2) - 4abcd
= a2(c2 - d2) - b2(c2 - d2) - 4abcd
= a2c2 - a2d2 - b2c2 + b2d2 - 4abcd
= a2c2 + b2d2 - a2d2 - b2c2 - 2abcd - 2abcd
= a2c2 + b2d2 - 2abcd - a2d2 - b2c2 - 2abcd.
= a2c2 + b2d2 - 2abcd - (a2d2 + b2c2 + 2abcd).
We know that,
(a + b)2 = a2 + 2ab + b2
and
(a - b)2 = a2 - 2ab + b2
∴ a2c2 + b2d2 - 2abcd - (a2d2 + b2c2 + 2abcd) = (ac - bd)2 - (ad + bc)2.
Using identity,
a2 - b2 = (a - b)(a + b).
(ac - bd)2 - (ad + bc)2 = [ac - bd - (ad + bc)](ac - bd + ad + bc)
= (ac - bd - ad - bc)(ac - bd + ad + bc).
Hence, (a2 - b2)(c2 - d2) - 4abcd = (ac - bd - ad - bc)(ac - bd + ad + bc).
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