## 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.