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Write the expanded form of (6a + 2b + 7c) 2 .
36a² + 4b² + 49c² + 28ab + 28bc + 84ac
36a² + 4b² + 49c² + 24ab + 28bc + 94ac
36a² + 4b² + 49c² + 24ab + 23bc + 84ac
36a² + 4b² + 49c² + 24ab + 28bc + 84ac
36a² + 4b² + 49c² + 24ab + 28bc + 84ac
Calculate the squares first: 36a2,4b2,49c2. Then check the cross-terms: 2(6a)(2b)=24ab, 2(2b)(7c)=28bc, 2(6a)(7c)=84ac. Match these with the options.
Expression to be expanded: (6a + 2b + 7c)2
(x+y+z)2=x2+y2+z2+2xy+2yz+2zx
Calculate the squares first: 36a2,4b2,49c2. Then check the cross-terms: 2(6a)(2b)=24ab, 2(2b)(7c)=28bc, 2(6a)(7c)=84ac. Match these with the options.
A common mistake is failing to multiply the cross-terms by the factor of 2 or forgetting to square the numerical coefficients (e.g., writing 6a2 instead of 36a2).
Identify terms
Compare the given expression (6a+2b+7c)2 with the identity (x+y+z)2. Here, x=6a, y=2b, and z=7c.
x=6a,y=2b,z=7c
Apply expansion formula
Substitute the values into the algebraic identity (x+y+z)2=x2+y2+z2+2xy+2yz+2zx.
(6a)2+(2b)2+(7c)2+2(6a)(2b)+2(2b)(7c)+2(6a)(7c)
Calculate squares and products
Square each term: (6a)2=36a2, (2b)2=4b2, and (7c)2=49c2. Calculate double products: 2(12ab)=24ab, 2(14bc)=28bc, and 2(42ac)=84ac.
36a2+4b2+49c2+24ab+28bc+84ac
D is correct because the expansion of (6a+2b+7c)2 results in 36a2+4b2+49c2+24ab+28bc+84ac.
This identity is fundamental to solving problems in Coordinate Geometry involving the distance formula or circle equations in general form.
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