(3a+4b+5c)^2

Jasonbradley

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Jun 16, 2024

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Expanding (3a + 4b + 5c)^2

The expression (3a + 4b + 5c)^2 represents the square of a trinomial. To expand it, we can apply the distributive property or use a specific formula.

Using the Distributive Property

1. Rewrite the expression: (3a + 4b + 5c)^2 = (3a + 4b + 5c)(3a + 4b + 5c)
2. Distribute: Multiply each term in the first trinomial by each term in the second trinomial.
   • (3a)(3a) + (3a)(4b) + (3a)(5c) + (4b)(3a) + (4b)(4b) + (4b)(5c) + (5c)(3a) + (5c)(4b) + (5c)(5c)
3. Simplify: Combine like terms.
   • 9a^2 + 12ab + 15ac + 12ab + 16b^2 + 20bc + 15ac + 20bc + 25c^2
   • Final result: 9a^2 + 24ab + 30ac + 16b^2 + 40bc + 25c^2

Using the Formula

The square of a trinomial can be expanded using the following formula:

(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

Applying this formula to our expression:

(3a + 4b + 5c)^2 = (3a)^2 + (4b)^2 + (5c)^2 + 2(3a)(4b) + 2(3a)(5c) + 2(4b)(5c)

Simplifying:

(3a + 4b + 5c)^2 = 9a^2 + 16b^2 + 25c^2 + 24ab + 30ac + 40bc

Therefore, the expanded form of (3a + 4b + 5c)^2 is 9a^2 + 24ab + 30ac + 16b^2 + 40bc + 25c^2.