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Simplify:(i) (a + b + c)2 + (a − b + c)2 (ii) (a + b + c)2 − (a − b + c)2 (iii) (a + b + c)2 + (a – b + c)2 + (a + b − c)2 (iv) (2x + p − c)2 − (2x − p + c)2 (v) (x2 + y2 − z2)2 − (x2 − y2 + z2)2 |
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Answer» (i) (a + b + c)2 + (a − b + c)2 = (a2 + b2 + c2 + 2ab+2bc+2ca) + (a2 + (−b)2 + c2 −2ab−2bc+2ca) = 2a2 + 2b2 + 2c2 + 4ca (ii) (a + b + c)2 − (a − b + c)2 = (a2 + b2 + c2 + 2ab + 2bc + 2ca) − (a2 + (−b)2 + c2 − 2ab − 2bc + 2ca) = a2 + b2 + c2 + 2ab + 2bc + 2ca − a2 − b2 − c2 + 2ab + 2bc − 2ca = 4ab + 4bc (iii) (a + b + c)2 + (a – b + c)2 + (a + b − c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca + (a2 + b2 + (c)2 − 2bc − 2cb + 2ca) + (a2 + b2 + c2 + 2ab − 2bc – 2ab) = 3a2 + 3b2 + 3c2 + 2ab − 2bc + 2ca (iv) (2x + p − c)2 − (2x − p + c)2 = [4x2 + p2 + c2 + 4xp − 2pc − 4xc] − [4x2 + p2 + c2 − 4xp− 2pc + 4xc] = 4x2 + p2 + c2 + 4xp − 2pc − 4cx − 4x2 − p2 − c2 + 4xp + 2pc− 4cx = 8xp − 8xc = 8(xp − xc) (v) (x2 + y2 − z2)2 − (x2 − y2 + z2)2 = (x2 + y2 + (−z)2)2 − (x2 − y2 + z2)2 = [x4 + y4 + z4 + 2x2y2 + 2y2z2 + 2x2z2 − [x4 + y4 + z4 − 2x2y2 − 2y2z2 + 2x2z2] = 4x2y2 – 4z2x2 |
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