(a+b+c)^2 Formula Proof

Suzhoumeite

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Jul 03, 2024

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(a+b+c)² Formula Proof

In this article, we will prove the formula for the square of a trinomial, which is widely used in algebra and mathematics. The formula is:

(a+b+c)² = a² + 2ab + 2ac + b² + 2bc + c²

Proof

Let's start by expanding the left-hand side of the equation:

(a+b+c)² = (a+b+c)(a+b+c)

Using the distributive property, we can expand the product of the two trinomials:

(a+b+c)(a+b+c) = a(a+b+c) + b(a+b+c) + c(a+b+c)

Now, let's expand each term:

a(a+b+c) = a² + ab + ac b(a+b+c) = ab + b² + bc c(a+b+c) = ac + bc + c²

Now, let's combine like terms:

a² + ab + ac + ab + b² + bc + ac + bc + c²

Combine the terms with the same variables:

a² + 2ab + 2ac + b² + 2bc + c²

And that's it! We have successfully proved the formula for the square of a trinomial.

Conclusion

The formula (a+b+c)² = a² + 2ab + 2ac + b² + 2bc + c² is a fundamental identity in algebra, and it has many applications in mathematics, physics, and engineering. By expanding the product of two trinomials, we have proved this formula and shown how it can be used to simplify complex expressions.