(a+b+c)^2 All Formula

Jasonbradley

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Jun 16, 2024

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The (a + b + c)^2 Formula Explained

The formula (a + b + c)^2 is a fundamental concept in algebra used to expand the square of a trinomial. Understanding this formula is crucial for simplifying expressions and solving equations. Let's break down the formula and explore its applications.

The Formula

The formula for (a + b + c)^2 is:

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

This formula states that the square of the sum of three terms is equal to the sum of the squares of each term plus twice the product of each pair of terms.

Derivation

The formula can be derived using the distributive property of multiplication:

1. Start with (a + b + c)^2: This means multiplying (a + b + c) by itself.
2. Expand the multiplication: (a + b + c) * (a + b + c)
3. Apply the distributive property: This involves multiplying each term in the first set of parentheses by each term in the second set of parentheses.
   • a * (a + b + c) = a^2 + ab + ac
   • b * (a + b + c) = ab + b^2 + bc
   • c * (a + b + c) = ac + bc + c^2
4. Combine like terms: a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

Applications

The (a + b + c)^2 formula has numerous applications in various areas of mathematics, including:

• Simplifying algebraic expressions: It can be used to simplify expressions involving trinomials raised to the power of two.
• Solving equations: It can be employed to solve quadratic equations with three variables.
• Geometry: It can be applied to find the area of certain geometric shapes.
• Calculus: It is used in derivative calculations involving functions of three variables.

Examples

Let's look at a few examples to illustrate the formula's usage:

1. Simplify (2x + 3y + 5z)^2:

   • Using the formula, we get: (2x + 3y + 5z)^2 = (2x)^2 + (3y)^2 + (5z)^2 + 2(2x)(3y) + 2(2x)(5z) + 2(3y)(5z)
   • Simplifying further: 4x^2 + 9y^2 + 25z^2 + 12xy + 20xz + 30yz

2. Find the value of (x + y + 1)^2 if x = 2 and y = 3:

   • Substitute the values of x and y into the formula: (2 + 3 + 1)^2 = (6)^2 = 36

Conclusion

The (a + b + c)^2 formula is a powerful tool for expanding and simplifying trinomials. Its applications extend beyond basic algebra, making it a fundamental concept in various branches of mathematics. Understanding this formula allows you to manipulate and solve complex equations and expressions more efficiently.