Sure, let's solve the given expression step-by-step.
We need to find the value of [tex]\((a - b - c)^2 - (a - b + c)^2\)[/tex].
### Step 1: Expand Each Square
First, let's expand each term separately.
For [tex]\((a - b - c)^2\)[/tex]:
[tex]\[
(a - b - c)^2 = (a - b - c)(a - b - c)
\][/tex]
Using the distributive property (also known as the FOIL method):
[tex]\[
= (a - b - c)(a - b - c) = a(a - b - c) - b(a - b - c) - c(a - b - c)
= a^2 - ab - ac - ab + b^2 + bc - ac + bc + c^2
\][/tex]
Combine like terms:
[tex]\[
(a - b - c)^2 = a^2 - 2ab + b^2 - 2ac + 2bc + c^2
\][/tex]
For [tex]\((a - b + c)^2\)[/tex]:
[tex]\[
(a - b + c)^2 = (a - b + c)(a - b + c)
\][/tex]
Again, using the distributive property:
[tex]\[
= (a - b + c)(a - b + c) = a(a - b + c) - b(a - b + c) + c(a - b + c)
= a^2 - ab + ac - ab + b^2 - bc + ac - bc + c^2
\][/tex]
Combine like terms:
[tex]\[
(a - b + c)^2 = a^2 - 2ab + b^2 + 2ac - 2bc + c^2
\][/tex]
### Step 2: Subtract the Two Expanded Expressions
We need to find the difference between these two expanded forms:
[tex]\[
(a - b - c)^2 - (a - b + c)^2
\][/tex]
Using the expanded results:
[tex]\[
= \left(a^2 - 2ab + b^2 - 2ac + 2bc + c^2\right) - \left(a^2 - 2ab + b^2 + 2ac - 2bc + c^2\right)
\][/tex]
Distribute the subtraction:
[tex]\[
= a^2 - 2ab + b^2 - 2ac + 2bc + c^2 - a^2 + 2ab - b^2 - 2ac + 2bc - c^2
\][/tex]
### Step 3: Simplify the Expression
Combine like terms by simplifying:
[tex]\[
= a^2 - a^2 - 2ab + 2ab + b^2 - b^2 - 2ac - 2ac + 2bc + 2bc + c^2 - c^2
= 0 - 4ac + 4bc
\][/tex]
So, the simplified form of the expression is:
[tex]\[
4c(-a + b)
\][/tex]
Thus,
[tex]\[
(a - b - c)^2 - (a - b + c)^2 = 4c(-a + b)
\][/tex]
This is the final result.
