2-formula-proof.jpeg "(a+b+c)2 Formula Proof")
**(a + b + c)**² Formula Proof
In algebra, the formula for the square of a sum of three terms, (a + b + c)², is a crucial concept in various mathematical operations. The formula is widely used in different branches of mathematics, physics, and engineering. In this article, we will provide a proof of the (a + b + c)² formula.
The Formula
The formula for the square of a sum of three terms is:
**(a + b + c)**² = a² + b² + c² + 2ab + 2bc + 2ca
Proof
To prove the formula, we can start by expanding the left-hand side of the equation:
**(a + b + c)**² = (a + b + c)(a + b + c)
Using the distributive property of multiplication over addition, we can expand the product as follows:
**(a + b + c)**² = a(a + b + c) + b(a + b + c) + c(a + b + c)
Now, let's expand each term:
=a² + ab + ac + ba + b² + bc + ca + cb + c²
Next, we can combine like terms:
=a² + b² + c² + 2ab + 2bc + 2ca
Thus, we have proven the formula:
**(a + b + c)**² = a² + b² + c² + 2ab + 2bc + 2ca
Importance and Applications
The (a + b + c)² formula has numerous applications in various fields, including:
- Algebra: The formula is used to simplify expressions involving the square of a sum of three terms.
- Geometry: The formula is used to calculate the area of a rectangle with sides a, b, and c.
- Trigonometry: The formula is used to simplify trigonometric identities involving the sum of three angles.
- Physics: The formula is used to calculate the energy of a system with three components.
In conclusion, the (a + b + c)² formula is a fundamental concept in algebra and has numerous applications in various fields. We have provided a proof of the formula, and it is essential to understand and remember it for solving various mathematical problems.
