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Write the expanded form of (5a + 4b + 9c) 2.
25a2+16b2+81c2+44ab+72bc+90ac
25a2+16b2+81c2+40ab+72bc+90ac
25a2+16b2+81c2+40ab+72bc+100ac
25a2+16b2+81c2+40ab+67bc+90ac
25a2+16b2+81c2+40ab+72bc+90ac
Use the formula by squaring each term (25, 16, 81) and then doubling the product of each pair (254=40, 249=72, 259=90).
Expression to expand: (5a + 4b + 9c)²
(x+y+z)2=x2+y2+z2+2xy+2yz+2zx
Use the formula by squaring each term (25, 16, 81) and then doubling the product of each pair (254=40, 249=72, 259=90).
Many students forget to multiply the cross-products by 2, or miscalculate the coefficient product of 5a and 9c (which is 45, doubled to 90).
Identify terms
Identify the components x=5a, y=4b, and z=9c from the expression (5a+4b+9c)2.
x=5a,y=4b,z=9c
Apply expansion formula
Substitute the terms into the expansion formula (x+y+z)2=x2+y2+z2+2xy+2yz+2zx.
(5a+4b+9c)2=(5a)2+(4b)2+(9c)2+2(5a)(4b)+2(4b)(9c)+2(5a)(9c)
Calculate squares and products
Calculate individual squares (25a2,16b2,81c2) and the doubled cross-products (40ab,72bc,90ac).
25a2+16b2+81c2+40ab+72bc+90ac
B is correct because the expansion of (5a+4b+9c)2 using the algebraic identity results in 25a2+16b2+81c2+40ab+72bc+90ac.
This identity is a foundational building block for solving quadratic equations and factoring complex trinomials in higher algebra.
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