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Express the expression below as a perfect square: a2+9b2+c2тИТ6ab+6bcтИТ2ac
(aтИТ3bтИТ2c)2
(a+3b+c)2
(aтИТ3bтИТc)2
(aтИТ3b+c)2
(aтИТ3bтИТc)2
Identify the squares of terms (a2, (3b)2, (2c)2 or c2) and check the signs of the cross-product terms to determine negative signs.
Expression: a┬▓ + 9b┬▓ + c┬▓ - 6ab + 6bc - 2ac
(x+y+z)2=x2+y2+z2+2xy+2yz+2zx
Identify the squares of terms (a2, (3b)2, (2c)2 or c2) and check the signs of the cross-product terms to determine negative signs.
Incorrectly assigning negative signs to the variables based on only one negative cross-product term.
Step 1: Identify the components
The expression contains three squared terms: a2, (3b)2, and c2 (or (тИТc)2).
a2+(3b)2+c2
Step 2: Analyze cross-product terms
Looking at the mixed terms: тИТ6ab, +6bc, and тИТ2ac. Since both тИТ6ab and тИТ2ac contain variable a and have negative signs, a must have the opposite sign to b and c, or equivalently, let us test a, тИТ3b, and тИТc.
тИТ6ab+6bcтИТ2ac
Step 3: Match with expansion formula
Consider the expansion of (aтИТ3bтИТc)2: (aтИТ3bтИТc)2=a2+(тИТ3b)2+(тИТc)2+2(a)(тИТ3b)+2(тИТ3b)(тИТc)+2(тИТc)(a)
=a2+9b2+c2тИТ6ab+6bcтИТ2ac
C is correct because expanding (aтИТ3bтИТc)2 gives the exact given expression a2+9b2+c2тИТ6ab+6bcтИТ2ac.
Trinomial expansion formulas are extensively used in polynomial factorization and completing the square for conic sections in coordinate geometry.
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