## The (a - b)^2 Formula: A Comprehensive Guide

The (a - b)^2 formula, also known as the **difference of squares formula**, is a fundamental concept in algebra that allows you to quickly expand expressions involving the square of a difference. This formula has wide applications in various mathematical fields and is essential for simplifying expressions and solving equations.

### Understanding the Formula

The (a - b)^2 formula states that:

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

This formula essentially tells us that the square of the difference of two terms is equal to the sum of the square of the first term, minus twice the product of the two terms, plus the square of the second term.

### How the Formula Works

The (a - b)^2 formula can be derived by expanding the square:

**(a - b)^2 = (a - b)(a - b)**

Applying the distributive property, we get:

**(a - b)(a - b) = a(a - b) - b(a - b)**

Expanding further:

**a(a - b) - b(a - b) = a^2 - ab - ba + b^2**

Simplifying by combining like terms:

**a^2 - ab - ba + b^2 = a^2 - 2ab + b^2**

Therefore, we arrive at the (a - b)^2 formula:

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

### Examples of Using the Formula

Here are some examples of how to apply the (a - b)^2 formula:

**Example 1:**

Expand (x - 3)^2

Using the formula, we have:

**(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9**

**Example 2:**

Simplify (2y - 5)^2

Applying the formula:

**(2y - 5)^2 = (2y)^2 - 2(2y)(5) + 5^2 = 4y^2 - 20y + 25**

**Example 3:**

Solve for x in the equation (x - 2)^2 = 16

Expanding the left side:

**x^2 - 4x + 4 = 16**

Subtracting 16 from both sides:

**x^2 - 4x - 12 = 0**

Factoring the quadratic equation:

**(x - 6)(x + 2) = 0**

Therefore, the solutions are:

**x = 6** or **x = -2**

### Applications of the (a - b)^2 Formula

The (a - b)^2 formula has various applications in different areas of mathematics, including:

**Simplifying algebraic expressions:**By using the formula, you can simplify complex expressions involving squares of differences.**Solving quadratic equations:**The formula is helpful in factoring quadratic equations and finding their solutions.**Geometry:**The formula can be used to find the area of squares and other geometric figures.**Calculus:**The formula can be used in derivative calculations and other operations in calculus.

### Conclusion

The (a - b)^2 formula is a valuable tool in algebra that simplifies calculations and expands expressions involving squares of differences. Mastering this formula is crucial for understanding and solving various mathematical problems across different fields.