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

3 min read Jun 16, 2024
(a+b+c+d)^2+(a+b-c-d)^2+(ac Bd)^2+(a+d-b-c)^2

Expanding and Simplifying (a+b+c+d)^2+(a+b-c-d)^2+(ac bd)^2+(a+d-b-c)^2

This expression involves squaring various sums and differences of variables, which can be expanded using the algebraic identity:

(x + y)^2 = x^2 + 2xy + y^2

Let's break down the simplification process step by step:

Step 1: Expanding the Squares

  • (a + b + c + d)^2:
    • This expands as: a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd
  • (a + b - c - d)^2:
    • This expands as: a^2 + b^2 + c^2 + d^2 + 2ab - 2ac - 2ad - 2bc + 2bd - 2cd
  • (ac bd)^2:
    • This is simply: a^2c^2 - 2abcd + b^2d^2
  • (a + d - b - c)^2:
    • This expands as: a^2 + b^2 + c^2 + d^2 - 2ab - 2ac + 2ad + 2bc - 2bd - 2cd

Step 2: Combining Like Terms

Now we combine all the terms from the expanded expressions:

  • a^2 terms: 4a^2
  • b^2 terms: 4b^2
  • c^2 terms: 4c^2
  • d^2 terms: 4d^2
  • ab terms: 2ab - 2ab = 0
  • ac terms: -2ac - 2ac = -4ac
  • ad terms: 2ad + 2ad = 4ad
  • bc terms: -2bc + 2bc = 0
  • bd terms: 2bd - 2bd = 0
  • cd terms: -2cd - 2cd = -4cd
  • abcd terms: -2abcd

Step 3: Final Simplification

Adding all the combined terms together, we get the simplified expression:

**(a + b + c + d)^2 + (a + b - c - d)^2 + (ac bd)^2 + (a + d - b - c)^2 = ** 4a^2 + 4b^2 + 4c^2 + 4d^2 - 4ac + 4ad - 4cd - 2abcd

This simplified expression represents the final form of the original expression after expansion and simplification.

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