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Write the expanded form of (8a + 3b + 9c) 2 .
64a 2 + 9b 2 + 81c 2 + 48ab + 54bc + 144ac
64a 2 + 9b 2 + 81c 2 + 52ab + 54bc + 144ac
64a 2 + 9b 2 + 81c 2 + 48ab + 54bc + 154ac
64a 2 + 9b 2 + 81c 2 + 48ab + 49bc + 144ac
64a 2 + 9b 2 + 81c 2 + 48ab + 54bc + 144ac
The expanded form of the algebraic expression (8a + 3b + 9c)2 is found using the algebraic identity for the square of a trinomial: (x + y + z)2 = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x = 8a, y = 3b, and z = 9c gives (8a)2 + (3b)2 + (9c)2 + 2(8a)(3b) + 2(3b)(9c) + 2(9c)(8a), which simplifies to 64a² + 9b² + 81c² + 48ab + 54bc + 144ac.
The expanded form of the algebraic expression (8a + 3b + 9c)2 is found using the algebraic identity for the square of a trinomial: (x + y + z)2 = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x = 8a, y = 3b, and z = 9c gives (8a)2 + (3b)2 + (9c)2 + 2(8a)(3b) + 2(3b)(9c) + 2(9c)(8a), which simplifies to 64a² + 9b² + 81c² + 48ab + 54bc + 144ac.
Algebraic expansion formulas form the foundational basis of quantitative aptitude in Indian competitive exams such as SSC CGL, CHSL, RRB NTPC, and banking examinations.
The square of a trinomial expansion always yields six terms: three squared terms and three double-product cross terms.
Order of variables in cross terms follows cyclic symmetry: ab, bc, and ac.
Coefficient calculations require careful handling of numerical coefficients multiplied by 2.
Option A correctly calculates all coefficients: 8² = 64, 3² = 9, 9² = 81, 283 = 48, 239 = 54, and 298 = 144.
Option B contains an incorrect coefficient of 52ab instead of 48ab.
Option C incorrectly lists 154ac instead of 144ac.
Option D incorrectly lists 49bc instead of 54bc.
A is correct — 64a 2 + 9b 2 + 81c 2 + 48ab + 54bc + 144ac is the exact expanded form using the trinomial square identity.
Mastering algebraic identities like (x+y+z)2 and (x+y)3 is essential for solving complex simplification and factorization questions in quantitative aptitude.
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