%5E2-formula-expansion.jpeg "(a-b-c)^2 Formula Expansion")
Understanding the (a-b-c)^2 Formula Expansion
The formula (a-b-c)^2 represents the square of a trinomial, where 'a', 'b', and 'c' can be any real numbers. Expanding this formula gives us a useful way to express the square of a trinomial in terms of individual terms.
The Expanded Form
The expansion of (a-b-c)^2 is:
** (a - b - c)^2 = a^2 + b^2 + c^2 - 2ab + 2ac - 2bc**
Derivation of the Formula
We can derive this formula using the following steps:
- Rewrite the expression: (a - b - c)^2 = (a - (b + c))^2
- Apply the square of a binomial formula: (a - (b + c))^2 = a^2 - 2a(b + c) + (b + c)^2
- Expand the remaining terms: a^2 - 2ab - 2ac + b^2 + 2bc + c^2
- Rearrange terms: a^2 + b^2 + c^2 - 2ab + 2ac - 2bc
Key Points to Remember
- The expansion of (a-b-c)^2 involves squaring each term of the trinomial.
- Cross-product terms are multiplied by -2.
- The formula holds true for any real numbers 'a', 'b', and 'c'.
Applications of the Formula
This formula has various applications in different areas of mathematics, including:
- Algebraic simplification: Expanding the square of a trinomial can simplify complex expressions.
- Solving equations: The formula can be used to solve equations involving squared trinomials.
- Geometry: The formula can be used to derive area formulas for certain geometric shapes.
- Physics: The formula can be applied in calculations involving vector quantities.
Example
Let's say we want to expand (x - 2y - 3)^2. Using the formula, we get:
(x - 2y - 3)^2 = x^2 + (2y)^2 + 3^2 - 2(x)(2y) + 2(x)(3) - 2(2y)(3)
Simplifying, we get:
x^2 + 4y^2 + 9 - 4xy + 6x - 12y
Conclusion
Understanding the (a-b-c)^2 formula expansion provides a powerful tool for working with trinomials. By applying this formula, we can simplify complex expressions and solve various problems in mathematics and other fields.