Simplifying the Expression: (a+b)(c-d)+(a-b)(c+d)+2(ac+bd)
This article will guide you through simplifying the given algebraic expression: (a+b)(c-d)+(a-b)(c+d)+2(ac+bd). We'll use the distributive property and some algebraic manipulation to arrive at the simplified form.
Expanding the Expressions
Let's start by expanding the first two terms using the distributive property (also known as FOIL):
- (a+b)(c-d) = ac - ad + bc - bd
- (a-b)(c+d) = ac + ad - bc - bd
Now, let's substitute these expanded forms back into the original expression:
(ac - ad + bc - bd) + (ac + ad - bc - bd) + 2(ac + bd)
Combining Like Terms
Next, we combine the like terms:
- ac + ac + 2ac = 4ac
- -ad + ad = 0
- bc - bc = 0
- -bd - bd + 2bd = 0
Simplified Form
After combining the terms, we are left with:
4ac
Therefore, the simplified form of the expression (a+b)(c-d)+(a-b)(c+d)+2(ac+bd) is 4ac.