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Write the expanded form of (6a + 8b + 3c) 2 .
36a 2 + 64b 2 + 9c 2 + 96ab + 43bc + 36ac
36a 2 + 64b 2 + 9c 2 + 100ab + 48bc + 36ac
36a 2 + 64b 2 + 9c 2 + 96ab + 48bc + 46ac
36a 2 + 64b 2 + 9c 2 + 96ab + 48bc + 36ac
36a 2 + 64b 2 + 9c 2 + 96ab + 48bc + 36ac
The expression (6a + 8b + 3c)² follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Applying this to 6a, 8b, and 3c, we calculate the squares as 36a², 64b², and 9c², and the doubled products as 2(6a)(8b) = 96ab, 2(8b)(3c) = 48bc, and 2(6a)(3c) = 36ac.
The expression (6a + 8b + 3c)² follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Applying this to 6a, 8b, and 3c, we calculate the squares as 36a², 64b², and 9c², and the doubled products as 2(6a)(8b) = 96ab, 2(8b)(3c) = 48bc, and 2(6a)(3c) = 36ac.
This problem relates to the algebraic identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca, a foundational concept in the NCERT Class 9 Mathematics curriculum.
The square of a trinomial includes the sum of the squares of each term.
The expansion contains three '2ab' type product terms.
Order of terms in the sum (6a, 8b, 3c) does not change the final expansion.
To expand (x+y+z)², square each individual variable: (6a)²=36a², (8b)²=64b², (3c)²=9c².
Cross-product verification: 2 * 6 * 8 = 96, 2 * 8 * 3 = 48, 2 * 6 * 3 = 36.
Option A is incorrect due to the faulty calculation of the bc and ac terms (43bc and 36ac vs 48bc and 36ac).
Option B is incorrect due to the coefficient of the ab term being 100 instead of 96.
D is correct — The expansion follows the identity (a+b+c)² = a²+b²+c²+2ab+2bc+2ca, resulting in 36a² + 64b² + 9c² + 96ab + 48bc + 36ac.
This identity is also essential in physics for calculating resultant vectors and magnitude of net forces in 3D coordinate systems.
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