Question
Write $(3a + 4b + 5c)^2$ in expanded form.
✓
Answer
Comparing the given expression with $(x + y + z)^2,$ we find that $x = 3a, y = 4b$ and $z = 5c.$
Therefore, using Identity $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx,$
we have $(3a + 4b + 5c)^2 = (3a)^2 + (4b)^2 + (5c)^2 + 2(3a)(4b) + 2(4b)(5c) + 2(5c)(3a)$
$= 9a^2 + 16b^2+ 25c^2+ 24ab + 40bc + 30ac$
Therefore, using Identity $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx,$
we have $(3a + 4b + 5c)^2 = (3a)^2 + (4b)^2 + (5c)^2 + 2(3a)(4b) + 2(4b)(5c) + 2(5c)(3a)$
$= 9a^2 + 16b^2+ 25c^2+ 24ab + 40bc + 30ac$
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