(a+b+c)2−(a−b−c)2

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying (a + b + c)² - (a - b - c)²

This expression involves squaring binomials and then subtracting the results. We can simplify it using the following steps:

1. Expanding the Squares:

We can expand the squares using the algebraic identity: (x + y)² = x² + 2xy + y².

• (a + b + c)²: Let's treat (b + c) as a single term.

  (a + b + c)² = a² + 2a(b + c) + (b + c)²

  Expanding further: a² + 2ab + 2ac + b² + 2bc + c²

• (a - b - c)²: Similarly, treating (-b - c) as a single term.

  (a - b - c)² = a² + 2a(-b - c) + (-b - c)²

  Expanding further: a² - 2ab - 2ac + b² + 2bc + c²

2. Subtracting the Expanded Expressions:

Now we substitute the expanded expressions back into the original expression:

(a + b + c)² - (a - b - c)² = (a² + 2ab + 2ac + b² + 2bc + c²) - (a² - 2ab - 2ac + b² + 2bc + c²)

3. Simplifying the Expression:

Notice that many terms cancel out:

• a² and -a² cancel out.
• b² and b² cancel out.
• c² and c² cancel out.

This leaves us with:

2ab + 2ac + 2ab + 2ac = 4ab + 4ac

Therefore, (a + b + c)² - (a - b - c)² simplifies to 4ab + 4ac.

Key Takeaway: This problem illustrates the power of using algebraic identities to simplify complex expressions. By strategically applying identities, we can often arrive at a simpler and more manageable form.