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

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

Expanding and Simplifying the Expression (a+b+c+d)^2+(a+b-c-d)^2+(a+c-b-d)^2+(a+d-b-c)^2

This expression represents the sum of squares of four different terms, each involving the variables 'a', 'b', 'c', and 'd'. Let's break down how to simplify this expression and uncover its underlying pattern.

Expanding the Squares

First, we need to expand each of the squares using the FOIL (First, Outer, Inner, Last) method or simply by applying the algebraic identity: (x + y)^2 = x^2 + 2xy + y^2

  • (a+b+c+d)^2: This expansion results in a large number of terms, but we can use a systematic approach to avoid missing any:

    • Square the first term: a^2
    • Multiply the first term by the second term and double the result: 2ab
    • Multiply the first term by the third term and double the result: 2ac
    • Multiply the first term by the fourth term and double the result: 2ad
    • Square the second term: b^2
    • Multiply the second term by the third term and double the result: 2bc
    • Multiply the second term by the fourth term and double the result: 2bd
    • Square the third term: c^2
    • Multiply the third term by the fourth term and double the result: 2cd
    • Square the fourth term: d^2
    • Combine all terms: a^2 + 2ab + 2ac + 2ad + b^2 + 2bc + 2bd + c^2 + 2cd + d^2

  • (a+b-c-d)^2: Following the same method as above, we get:

    • a^2 + 2ab - 2ac - 2ad + b^2 - 2bc - 2bd + c^2 + 2cd + d^2

  • (a+c-b-d)^2:

    • a^2 - 2ab + 2ac - 2ad + b^2 - 2bc + 2bd + c^2 - 2cd + d^2

  • (a+d-b-c)^2:

    • a^2 - 2ab - 2ac + 2ad + b^2 + 2bc - 2bd + c^2 + 2cd + d^2

Simplifying the Expression

Now, we can add all the expanded terms together. Notice that many terms cancel out due to their opposite signs:

  • a^2: (4)a^2
  • b^2: (4)b^2
  • c^2: (4)c^2
  • d^2: (4)d^2
  • 2ab: (2)2ab - (2)2ab = 0
  • 2ac: (2)2ac - (2)2ac = 0
  • 2ad: (2)2ad - (2)2ad = 0
  • 2bc: (2)2bc - (2)2bc = 0
  • 2bd: (2)2bd - (2)2bd = 0
  • 2cd: (2)2cd - (2)2cd = 0

The simplified expression is: 4a^2 + 4b^2 + 4c^2 + 4d^2

Final Result

We can further simplify by factoring out 4: 4(a^2 + b^2 + c^2 + d^2)

Therefore, the simplified form of the original expression is 4(a^2 + b^2 + c^2 + d^2). This demonstrates that the expression represents four times the sum of squares of the individual variables.

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