(a-b)^2=a^2-2ab+b^2 Proof

Jasonbradley

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Jun 16, 2024

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Proof of the Square of a Binomial: (a-b)² = a² - 2ab + b²

This proof demonstrates the expansion of the square of a binomial (a-b), which is frequently used in algebra and other mathematical fields.

Understanding the Concept

The square of a binomial (a-b)² means multiplying the binomial by itself:

(a - b)² = (a - b)(a - b)

Using the Distributive Property

We can expand this product using the distributive property:

1. Distribute the first term (a) of the first binomial to the second binomial: a(a - b) = a² - ab

2. Distribute the second term (-b) of the first binomial to the second binomial: -b(a - b) = -ab + b²

3. Combine the results from steps 1 and 2: a² - ab - ab + b²

Simplifying the Expression

Finally, combine the like terms:

a² - 2ab + b²

Therefore, we have proven that:

(a - b)² = a² - 2ab + b²

Visualizing the Proof

We can also visualize this proof using a geometric representation:

Imagine a square with sides of length (a - b). The area of this square is (a - b)². We can divide this square into four smaller regions:

1. A square with side length 'a' (area a²)
2. A rectangle with sides of length 'a' and 'b' (area ab)
3. Another rectangle with sides of length 'a' and 'b' (area ab)
4. A square with side length 'b' (area b²)

The total area of the large square is equal to the sum of the areas of the four smaller regions:

(a - b)² = a² - ab - ab + b² = a² - 2ab + b²

This visual representation further clarifies the proof and reinforces the concept.