(a+b-c+d)^2-(a-b+c-d)^2

2 min read Jun 16, 2024
(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.