(a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d)

Jasonbradley

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Jun 16, 2024

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Expanding and Simplifying Algebraic Expressions

This article explores the given algebraic equation:

(a + b + c + d)(a - b - c + d) = (a - b + c - d)(a + b - c - d)

We will prove that this equation holds true for any values of a, b, c, and d.

Expanding Both Sides

Let's begin by expanding both sides of the equation using the distributive property (also known as FOIL):

Left Side: (a + b + c + d)(a - b - c + d) = a(a - b - c + d) + b(a - b - c + d) + c(a - b - c + d) + d(a - b - c + d) = a² - ab - ac + ad + ab - b² - bc + bd + ac - bc - c² + cd + ad - bd - cd + d² = a² - b² - c² + d² + 2ad - 2bc

Right Side: (a - b + c - d)(a + b - c - d) = a(a + b - c - d) - b(a + b - c - d) + c(a + b - c - d) - d(a + b - c - d) = a² + ab - ac - ad - ab - b² + bc + bd + ac + bc - c² - cd - ad - bd + cd + d² = a² - b² - c² + d² + 2ad - 2bc

Conclusion

As we can see, both the left and right sides of the equation simplify to the same expression: a² - b² - c² + d² + 2ad - 2bc. Therefore, the equation is true for any values of a, b, c, and d.

This demonstrates a key principle in algebra: expanding and simplifying expressions can help to reveal underlying relationships and prove equations.