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Write the expanded form of (9a + 6b + 4c) 2 .
81a2+36b2+16c2+108ab+43bc+72ac
81a2+36b2+16c2+108ab+48bc+72ac
81a2+36b2+16c2+108ab+48bc+82ac
81a2+36b2+16c2+112ab+48bc+72ac
81a2+36b2+16c2+108ab+48bc+72ac
Focus on the cross-product terms: 2(9a)(6b) = 108ab, 2(6b)(4c) = 48bc, and 2(9a)(4c) = 72ac. Only option B matches these specific cross-product coefficients.
Expression to expand: (9a + 6b + 4c)2
(x+y+z)2=x2+y2+z2+2xy+2yz+2zx
Focus on the cross-product terms: 2(9a)(6b) = 108ab, 2(6b)(4c) = 48bc, and 2(9a)(4c) = 72ac. Only option B matches these specific cross-product coefficients.
Many students forget to multiply the coefficients by 2 when calculating the middle terms (2xy, 2yz, 2zx) or make sign errors with the variables.
Identify Terms
Identify x=9a, y=6b, and z=4c in the expansion formula (x+y+z)2=x2+y2+z2+2xy+2yz+2zx.
x=9a,y=6b,z=4c
Calculate Square Terms
Calculate the square of each individual term: (9a)2=81a2, (6b)2=36b2, and (4c)2=16c2.
81a2+36b2+16c2
Calculate Cross-Product Terms
Calculate the doubling cross-products: 2(9a)(6b)=108ab, 2(6b)(4c)=48bc, and 2(9a)(4c)=72ac.
108ab+48bc+72ac
Combine Results
Add all components to get the final expanded expression.
81a2+36b2+16c2+108ab+48bc+72ac
B is correct because the expansion of (9a+6b+4c)2 using the identity (x+y+z)2=x2+y2+z2+2xy+2yz+2zx results in 81a2+36b2+16c2+108ab+48bc+72ac.
This identity is fundamental for solving quadratic equations and finding the surface area of rectangular prisms in geometry exams.
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