## The Square of a Difference: Understanding (a - b)² = a² - 2ab + b²

The formula **(a - b)² = a² - 2ab + b²** is a fundamental algebraic identity that describes the expansion of the square of a difference. This formula is extremely useful in simplifying expressions, solving equations, and performing various algebraic operations.

### Understanding the Formula

The formula states that squaring a binomial of the form (a - b) results in the sum of the squares of the individual terms (a² and b²) minus twice the product of the two terms (2ab).

**Here's a breakdown:**

**(a - b)²:**This represents the square of the binomial (a - b).**a²:**This is the square of the first term (a).**b²:**This is the square of the second term (b).**-2ab:**This represents twice the product of the first and second terms (a and b).

### Proof of the Formula

We can prove the formula using the distributive property of multiplication:

(a - b)² = (a - b)(a - b)

Expanding this product, we get:

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

Therefore, we have successfully proven the formula **(a - b)² = a² - 2ab + b²**.

### Applications of the Formula

This formula has widespread applications in various mathematical fields:

**Simplifying expressions:**It allows us to expand squares of binomials, simplifying complex expressions.**Solving equations:**It helps in solving quadratic equations by factoring or completing the square.**Geometry:**It can be used to find the area of squares and other geometric figures.**Calculus:**It is used in deriving formulas for derivatives and integrals.

### Examples

Here are some examples of how to apply the formula:

**1. Expanding (x - 3)²:**

Using the formula, we get:

(x - 3)² = x² - 2(x)(3) + 3² = x² - 6x + 9

**2. Solving the equation (x - 2)² = 9:**

Expanding the left side using the formula:

x² - 4x + 4 = 9 x² - 4x - 5 = 0

Factoring the quadratic equation, we get:

(x - 5)(x + 1) = 0

Therefore, the solutions are x = 5 and x = -1.

### Conclusion

The formula **(a - b)² = a² - 2ab + b²** is a fundamental identity that simplifies algebraic expressions and has numerous applications in various mathematical fields. By understanding and applying this formula, we can effectively solve equations, simplify expressions, and gain a deeper understanding of algebraic concepts.