%5E2-(a-b%2Bc-d)%5E2.jpeg "(a+b-c+d)^2-(a-b+c-d)^2")
Expanding and Simplifying (a+b-c+d)^2-(a-b+c-d)^2
This article aims to simplify the expression (a+b-c+d)^2-(a-b+c-d)^2 by expanding it and combining like terms.
Expanding the Squares
We begin by expanding the squares using the formula (x+y)^2 = x^2 + 2xy + y^2. Applying this formula to both terms:
- (a+b-c+d)^2 = a^2 + 2ab - 2ac + 2ad + b^2 - 2bc + 2bd + c^2 - 2cd + d^2
- (a-b+c-d)^2 = a^2 - 2ab + 2ac - 2ad + b^2 - 2bc + 2bd + c^2 - 2cd + d^2
Combining Like Terms
Now, let's subtract the second expanded term from the first:
(a^2 + 2ab - 2ac + 2ad + b^2 - 2bc + 2bd + c^2 - 2cd + d^2) - (a^2 - 2ab + 2ac - 2ad + b^2 - 2bc + 2bd + c^2 - 2cd + d^2)
Notice that several terms cancel out:
- a^2 and -a^2
- b^2 and -b^2
- c^2 and -c^2
- d^2 and -d^2
- -2bc and -2bc
- 2bd and 2bd
This leaves us with:
2ab - 2ac + 2ad + 2ab - 2ac + 2ad
Final Simplification
Finally, combine the remaining like terms:
(2ab + 2ab) + (-2ac - 2ac) + (2ad + 2ad) = 4ab - 4ac + 4ad
Therefore, the simplified form of the expression (a+b-c+d)^2-(a-b+c-d)^2 is 4ab - 4ac + 4ad.