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Write the expanded form of (4a + 9b + 5c) 2 .
16a 2 + 81b 2 + 25c 2 + 76ab + 90bc + 40ac
16a 2 + 81b 2 + 25c 2 + 72ab + 90bc + 40ac
16a 2 + 81b 2 + 25c 2 + 72ab + 90bc + 50ac
16a 2 + 81b 2 + 25c 2 + 72ab + 85bc + 40ac
16a 2 + 81b 2 + 25c 2 + 72ab + 90bc + 40ac
The expanded form of the algebraic expression (4a + 9b + 5c) squared is obtained using the standard trinomial expansion identity: (x + y + z) squared = x squared + y squared + z squared + 2xy + 2yz + 2zx. Substituting x = 4a, y = 9b, and z = 5c yields 16a squared + 81b squared + 25c squared + 72ab + 90bc + 40ac, which corresponds directly to Option B.
The expanded form of the algebraic expression (4a + 9b + 5c) squared is obtained using the standard trinomial expansion identity: (x + y + z) squared = x squared + y squared + z squared + 2xy + 2yz + 2zx. Substituting x = 4a, y = 9b, and z = 5c yields 16a squared + 81b squared + 25c squared + 72ab + 90bc + 40ac, which corresponds directly to Option B.
Fundamental algebraic identities taught in secondary mathematics form the foundation for quantitative aptitude tested in all major Indian competitive examinations.
Trinomial square expansion involves squaring each individual term and adding twice the product of each distinct pair.
The coefficient of ab is 2 * 4 * 9 = 72.
The coefficient of bc is 2 * 9 * 5 = 90.
The coefficient of ac is 2 * 4 * 5 = 40.
Option A is incorrect because the ab term coefficient is incorrectly stated as 76 instead of 72.
Option C is incorrect because the ac term coefficient is incorrectly stated as 50 instead of 40.
Option D is incorrect because the bc term coefficient is incorrectly stated as 85 instead of 90.
Option B is correct тАФ 16a 2 + 81b 2 + 25c 2 + 72ab + 90bc + 40ac is the exact expanded form derived from the trinomial square identity.
Mastering algebraic identities drastically reduces calculation time in quantitative aptitude sections for quadratic equations and polynomial factorization problems.
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