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

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (a+b+c)^2-(b+c-a)^2+(c+a-b)^2-(a+b-c)^2

This article will explore the simplification of the expression: (a+b+c)^2-(b+c-a)^2+(c+a-b)^2-(a+b-c)^2. We will utilize algebraic manipulations to arrive at a simplified form.

Expanding the Squares

First, let's expand each of the squared terms using the formula (x + y)^2 = x^2 + 2xy + y^2:

• (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc
• (b+c-a)^2 = b^2 + c^2 + a^2 + 2bc - 2ab - 2ac
• (c+a-b)^2 = c^2 + a^2 + b^2 + 2ac - 2ab - 2bc
• (a+b-c)^2 = a^2 + b^2 + c^2 + 2ab - 2ac - 2bc

Substituting and Simplifying

Now, let's substitute these expanded forms back into the original expression:

(a^2 + b^2 + c^2 + 2ab + 2ac + 2bc) - (b^2 + c^2 + a^2 + 2bc - 2ab - 2ac) + (c^2 + a^2 + b^2 + 2ac - 2ab - 2bc) - (a^2 + b^2 + c^2 + 2ab - 2ac - 2bc)

Notice that many terms cancel out:

• a^2, b^2, c^2 cancel out
• 2bc cancels out
• -2ab cancels out
• -2ac cancels out

This leaves us with:

2ab + 2ac + 2bc + 2ab + 2ac + 2bc

Final Result

Combining like terms, we arrive at the simplified expression:

**(a+b+c)^2-(b+c-a)^2+(c+a-b)^2-(a+b-c)^2 = ** 4ab + 4ac + 4bc

Therefore, the original expression simplifies to 4ab + 4ac + 4bc.