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Write the expanded form of (5a + 7b + 3c) 2 .
25a 2 + 49b 2 + 9c 2 + 70ab + 37bc + 30ac
25a 2 + 49b 2 + 9c 2 + 70ab + 42bc + 30ac
25a 2 + 49b 2 + 9c 2 + 74ab + 42bc + 30ac
25a 2 + 49b 2 + 9c 2 + 70ab + 42bc + 40ac
25a 2 + 49b 2 + 9c 2 + 70ab + 42bc + 30ac
The expansion of the trinomial square (5a + 7b + 3c)² is solved using the standard algebraic identity: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x = 5a, y = 7b, and z = 3c gives (5a)² + (7b)² + (3c)² + 2(5a)(7b) + 2(7b)(3c) + 2(3c)(5a), which simplifies to 25a² + 49b² + 9c² + 70ab + 42bc + 30ac.
The expansion of the trinomial square (5a + 7b + 3c)² is solved using the standard algebraic identity: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. Substituting x = 5a, y = 7b, and z = 3c gives (5a)² + (7b)² + (3c)² + 2(5a)(7b) + 2(7b)(3c) + 2(3c)(5a), which simplifies to 25a² + 49b² + 9c² + 70ab + 42bc + 30ac.
Algebraic identities form the foundation of quantitative aptitude tested extensively in Indian competitive examinations like SSC CGL, CHSL, RRB NTPC, and banking exams.
The square of a trinomial with three terms always yields six terms in its expanded form.
Cross-terms are generated by multiplying each distinct pair of variables by 2.
Coefficient calculations require careful arithmetic: 2 × 5 × 7 = 70 for ab, 2 × 7 × 3 = 42 for bc, and 2 × 3 × 5 = 30 for ac.
Option B correctly calculates all individual squares and cross-product terms.
Option A contains an incorrect coefficient for the bc term (37bc instead of 42bc).
Options C and D contain incorrect coefficients in the ab and ac cross-terms.
B is correct — 25a 2 + 49b 2 + 9c 2 + 70ab + 42bc + 30ac is the exact expanded form derived from the algebraic identity (x + y + z)².
Mastering algebraic expansions like (a+b+c)² directly accelerates simplification problems in quantitative aptitude, reducing calculation time in competitive exams.
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