What is

The expression $a^2 – b^2$ is known as the difference of squares. This is a common algebraic identity that can be factored in a specific way.

Understanding the Difference of Squares

Definition

The difference of squares refers to the subtraction of one squared term from another squared term. Mathematically, it is written as:
$a^2 – b^2$

Factoring the Difference of Squares

One of the most useful properties of the difference of squares is that it can be factored into the product of two binomials. The formula for factoring $a^2 – b^2$ is:
$a^2 – b^2 = (a + b)(a – b)$

Proof of the Formula

Let’s verify this formula by expanding the right-hand side:

$(a + b)(a – b)$

Using the distributive property (also known as the FOIL method for binomials), we get:

$(a + b)(a – b) = a(a – b) + b(a – b)$

Expanding each term separately, we obtain:

$= a^2 – ab + ab – b^2$

Notice that the $-ab$ and $+ab$ terms cancel each other out, leaving us with:

$= a^2 – b^2$

This confirms that:

$a^2 – b^2 = (a + b)(a – b)$

Examples to Illustrate

Example 1

Let’s factor $9 – 4$

First, recognize that $9$ and $4$ are perfect squares:

$9 = 3^2$ and $4 = 2^2$

So, $9 – 4$ can be written as:

$3^2 – 2^2$

Using the difference of squares formula:

$3^2 – 2^2 = (3 + 2)(3 – 2) = 5 times 1 = 5$

Example 2

Factor $x^2 – 25$

Recognize that $x^2$ and $25$ are perfect squares:

$x^2$ and $25 = 5^2$

So, $x^2 – 25$ can be written as:

$x^2 – 5^2$

Using the difference of squares formula:

$x^2 – 5^2 = (x + 5)(x – 5)$

Example 3

Factor $4y^2 – 81$

Recognize that $4y^2$ and $81$ are perfect squares:

$4y^2 = (2y)^2$ and $81 = 9^2$

So, $4y^2 – 81$ can be written as:

$(2y)^2 – 9^2$

Using the difference of squares formula:

$(2y)^2 – 9^2 = (2y + 9)(2y – 9)$

Why is This Important?

Understanding how to factor the difference of squares is crucial for solving various algebraic problems. It simplifies expressions and equations, making them easier to work with. This technique is widely used in algebra, calculus, and even in real-world applications such as physics and engineering.

Conclusion

The difference of squares, $a^2 – b^2$, is a fundamental algebraic identity that can be factored into $(a + b)(a – b)$. This property is not only a powerful tool in simplifying mathematical expressions but also serves as a stepping stone for more advanced topics in mathematics.

Citations

1. 1. Khan Academy – Factoring
2. 2. Math is Fun – Factoring
3. 3. Purplemath – Factoring Polynomials