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Write the expanded form of (4a + 6b + 9c) 2 .
16a 2 + 36b 2 + 81c 2 + 48ab + 103bc + 72ac
16a 2 + 36b 2 + 81c 2 + 48ab + 108bc + 82ac
16a 2 + 36b 2 + 81c 2 + 52ab + 108bc + 72ac
16a 2 + 36b 2 + 81c 2 + 48ab + 108bc + 72ac
16a 2 + 36b 2 + 81c 2 + 48ab + 108bc + 72ac
The expression (4a + 6b + 9c)² is expanded using the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. By substituting x=4a, y=6b, and z=9c, we get (4a)² + (6b)² + (9c)² + 2(4a)(6b) + 2(6b)(9c) + 2(9c)(4a), which simplifies to 16a² + 36b² + 81c² + 48ab + 108bc + 72ac.
The expression (4a + 6b + 9c)² is expanded using the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. By substituting x=4a, y=6b, and z=9c, we get (4a)² + (6b)² + (9c)² + 2(4a)(6b) + 2(6b)(9c) + 2(9c)(4a), which simplifies to 16a² + 36b² + 81c² + 48ab + 108bc + 72ac.
This concept falls under Algebraic Identities, a fundamental topic in Quantitative Aptitude which is a permanent fixture in the SSC CGL, CHSL, and Railways (RRB) examination syllabi.
Algebraic identities are essential for solving polynomial expansion problems in SSC and Banking exams.
The sum of squares of individual terms and twice the sum of products of terms in pairs forms the expansion.
Mistakes often occur in signs or forgetting the '2' multiplier during multiplication of the double products.
Option D is the correct derivation resulting from the identity applied to the given variables.
Option A has an incorrect coefficient for the 'bc' term (103 instead of 108).
Option B has an incorrect coefficient for the 'ac' term (82 instead of 72).
Option C has an incorrect coefficient for the 'ab' term (52 instead of 48).
D is correct — 16a² + 36b² + 81c² + 48ab + 108bc + 72ac is the accurate expansion of the trinomial square.
Mastering algebraic identities will significantly speed up calculations in Quantitative Aptitude sections, specifically for chapters like Algebra, Simplification, and Geometry area-related formulas.
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