To solve for [tex]\((a - b - c)^2\)[/tex], we need to expand and simplify the expression step by step. Let's break it down:
1. Understanding the Squared Term:
- We start with [tex]\((a - b - c)^2\)[/tex].
- Using the algebraic identity [tex]\((x - y)^2 = x^2 - 2xy + y^2\)[/tex], we can decompose the given expression.
2. Rewriting the Expression:
- Rewrite [tex]\((a - b - c)^2\)[/tex] as [tex]\((a - (b + c))^2\)[/tex].
- Let [tex]\(x = a\)[/tex] and [tex]\(y = (b + c)\)[/tex], so we now have [tex]\((a - (b + c))\)[/tex].
3. Applying the Square Formula:
- Apply [tex]\((x - y)^2 = x^2 - 2xy + y^2\)[/tex]:
[tex]\[
(a - (b + c))^2 = a^2 - 2a(b + c) + (b + c)^2
\][/tex]
4. Expanding [tex]\((b + c)^2\)[/tex]:
- Expand the term [tex]\((b + c)^2\)[/tex] using the identity [tex]\((m + n)^2 = m^2 + 2mn + n^2\)[/tex]:
[tex]\[
(b + c)^2 = b^2 + 2bc + c^2
\][/tex]
5. Combining All Terms:
- Now, substitute [tex]\((b + c)^2\)[/tex] back into the original expanded form:
[tex]\[
(a - (b + c))^2 = a^2 - 2a(b + c) + b^2 + 2bc + c^2
\][/tex]
6. Distributing and Simplifying:
- Distribute [tex]\( -2a\)[/tex] through the term [tex]\((b + c)\)[/tex]:
[tex]\[
a^2 - 2ab - 2ac + b^2 + 2bc + c^2
\][/tex]
So, the fully expanded and simplified form of [tex]\((a - b - c)^2\)[/tex] is:
[tex]\[
(a - b - c)^2 = a^2 - 2ab - 2ac + b^2 + 2bc + c^2
\][/tex]
