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Write the expanded form of (6a + 5b + 8c) 2 .
36a 2 + 25b 2 + 64c 2 + 60ab + 80bc + 96ac
36a 2 + 25b 2 + 64c 2 + 60ab + 80bc + 106ac
36a 2 + 25b 2 + 64c 2 + 64ab + 80bc + 96ac
36a 2 + 25b 2 + 64c 2 + 60ab + 75bc + 96ac
36a 2 + 25b 2 + 64c 2 + 60ab + 80bc + 96ac
The expression (6a + 5b + 8c)² follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x=6a, y=5b, and z=8c results in 36a² + 25b² + 64c² + 2(6a)(5b) + 2(5b)(8c) + 2(8c)(6a), which simplifies to 36a² + 25b² + 64c² + 60ab + 80bc + 96ac.
The expression (6a + 5b + 8c)² follows the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x=6a, y=5b, and z=8c results in 36a² + 25b² + 64c² + 2(6a)(5b) + 2(5b)(8c) + 2(8c)(6a), which simplifies to 36a² + 25b² + 64c² + 60ab + 80bc + 96ac.
This algebraic identity is a fundamental property of binomial expansions and trinomial squares used in competitive mathematics exams to test basic proficiency in polynomials.
Expanding trinomial squares requires applying the identity (x+y+z)² = x² + y² + z² + 2xy + 2yz + 2zx.
The coefficient of the squared terms is obtained by squaring individual coefficients: 6²=36, 5²=25, 8²=64.
The product terms (ab, bc, ac) involve doubling the product of the numerical coefficients.
Common errors in exams involve missing the doubling (multiplication by 2) for the product terms.
Option B is incorrect because 106ac is a calculation error of the 286 product.
Option C is incorrect because 64ab is a calculation error of the 265 product.
Option D is incorrect because 75bc is a calculation error of the 258 product.
A is correct — The expansion follows the identity (6a + 5b + 8c)² = 36a² + 25b² + 64c² + 2(6a)(5b) + 2(5b)(8c) + 2(8c)(6a) = 36a² + 25b² + 64c² + 60ab + 80bc + 96ac.
Algebraic identities are frequently tested in Quantitative Aptitude sections. Memorizing (x+y+z)² and (x+y)³ helps in solving complex simplification questions in record time.
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