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Write the expanded form of (8a + 5b + 9c) 2 .
64a 2 + 25b 2 + 81c 2 + 84ab + 90bc + 144ac
64a 2 + 25b 2 + 81c 2 + 80ab + 85bc + 144ac
64a 2 + 25b 2 + 81c 2 + 80ab + 90bc + 144ac
64a 2 + 25b 2 + 81c 2 + 80ab + 90bc + 154ac
64a 2 + 25b 2 + 81c 2 + 80ab + 90bc + 144ac
The expression (8a + 5b + 9c)² follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. By substituting x=8a, y=5b, and z=9c, we calculate the squares as 64a², 25b², and 81c², and the cross-products as 80ab (2×8×5), 90bc (2×5×9), and 144ac (2×8×9).
The expression (8a + 5b + 9c)² follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. By substituting x=8a, y=5b, and z=9c, we calculate the squares as 64a², 25b², and 81c², and the cross-products as 80ab (2×8×5), 90bc (2×5×9), and 144ac (2×8×9).
This problem relates to fundamental Algebraic Identities, which form the bedrock of quantitative aptitude in all Indian competitive exams, including UPSC CSAT, SSC CGL, and Banking examinations.
The identity (x+y+z)² always results in three squared terms and three distinct cross-product terms.
Order of operations: First calculate the squares of coefficients, then the product of 2 multiplied by each pair of variables.
Attention to detail: The cross-product terms are 2xy, 2yz, and 2zx respectively.
Cross-product 2xy: 2 × 8 × 5 = 80; 2yz: 2 × 5 × 9 = 90; 2zx: 2 × 8 × 9 = 144.
Option A is incorrect due to calculation errors in the cross-product terms (84ab vs 80ab).
Option B and D are incorrect because they provide incorrect coefficients for the cross-products or final terms.
C is correct — it accurately applies the algebraic identity (x+y+z)² to the given terms (8a + 5b + 9c)².
Mastering these expansions is essential for simplifying complex equations in Physics (Kinematics formulas) and Finance (Compound interest growth) commonly found in competitive papers.
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