Factoring is a fundamental skill in algebra, and one of the most common patterns you'll encounter is the **difference of two squares**. This powerful algebraic identity simplifies complex expressions and is a shortcut for a variety of mathematical problems. This guide will walk you through the formula, provide clear examples, and show you why this method is so useful.
1. What is the Difference of Two Squares?
A "difference of two squares" is any expression that can be written in the form a² - b². Here, a² and b² are perfect squares, and "difference" means subtraction. The key is to recognize that you are subtracting one perfect square term from another.
Examples of Difference of Squares:
x² - 9
16 - y²
4x² - 25
100 - z²2. The Factoring Formula
The formula for factoring the difference of two squares is:
Formula:
a² - b² = (a - b)(a + b)This identity shows that the difference of two squares can be factored into two binomials: one with a subtraction sign and one with an addition sign.
Verification:
You can easily prove this by multiplying the right side using the FOIL (First, Outer, Inner, Last) method:
(a - b)(a + b) = a·a + a·b - b·a - b·b
= a² + ab - ab - b²
= a² - b²The middle terms, +ab and -ab, cancel each other out, leaving you with the original expression.
3. Step-by-Step Examples
Method:
- Identify the first term and find its square root (a).
- Identify the second term and find its square root (b).
- Write the factors as (a - b)(a + b).
Example 1: Factor x² - 49
Step 1: The square root of x² is x (a = x).
Step 2: The square root of 49 is 7 (b = 7).
Step 3: Write the factors → (x - 7)(x + 7).
Answer: (x - 7)(x + 7)Example 2: Factor 9y² - 16
Step 1: The square root of 9y² is 3y (a = 3y).
Step 2: The square root of 16 is 4 (b = 4).
Step 3: Write the factors → (3y - 4)(3y + 4).
Answer: (3y - 4)(3y + 4)4. Real-World Applications & Mental Math
The difference of squares formula isn't just for abstract algebra problems. It's a great tool for mental math and simplifying arithmetic.
Example: Quick Calculation of 52² - 48²
This looks complicated, but using the formula:
52² - 48² = (52 - 48)(52 + 48)
= (4)(100)
= 400This method is far easier and faster than squaring both numbers and then subtracting.
5. Practice Problems
Test your understanding with these problems:
1. Factor x² - 100
2. Factor 4a² - 81b²
3. Simplify 15² - 5²
6. Conclusion
The difference of two squares is one of the most useful and elegant identities in algebra. Recognizing this pattern can simplify complex expressions, speed up calculations, and lay the groundwork for more advanced factoring techniques. Consistent practice is the key to making this shortcut an intuitive part of your mathematical toolkit.
Want to improve your factoring skills?
Check out our Algebra Tips & Tricks section for more helpful guides!