Difference of Squares- Factoring Made Simple
What Is the Difference of Squares?
The difference of squares is a factoring pattern you need to memorize. It works every time, no exceptions.
The formula is simple:
a² - b² = (a + b)(a - b)
That's it. One expression factors into two binomials. The left side shows subtraction between two perfect squares. The right side shows the same bases added and subtracted from each other.
This pattern shows up constantly in algebra, so you better know it cold.
Why This Formula Works
Multiplying out the right side proves it:
(a + b)(a - b) = a² - ab + ab - b² = a² - b²
The middle terms cancel. You get exactly what you started with. The formula is reversible, which is why it works for factoring.
How to Spot a Difference of Squares
Your expression must meet two conditions:
- Two terms connected by subtraction (minus sign)
- Both terms are perfect squares
If either condition fails, this pattern won't help you. Check for perfect squares first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on.
Factoring Examples
Example 1: x² - 9
x² is a perfect square (x times x). 9 is a perfect square (3 times 3). The minus sign connects them.
This fits the pattern. Apply the formula:
x² - 9 = (x + 3)(x - 3)
Done. Two seconds max.
Example 2: 4a² - 25
4a² is (2a)². 25 is 5². Subtract them.
4a² - 25 = (2a + 5)(2a - 5)
Example 3: 49y² - 16
49y² is (7y)². 16 is 4².
49y² - 16 = (7y + 4)(7y - 4)
Example 4: x⁶ - 1
This one trips people up. x⁶ is (x³)². 1 is 1².
x⁶ - 1 = (x³ + 1)(x³ - 1)
Notice you might need to factor further. x³ - 1 and x³ + 1 are cubes, not this pattern. Keep going if the problem asks for complete factorization.
Getting Started: Step-by-Step Process
Here's how to factor any difference of squares problem:
- Confirm it's subtraction — if you see addition, stop here. This pattern doesn't apply.
- Verify both terms are perfect squares — take square roots mentally. If you get non-integers with no variable, this isn't your pattern.
- Identify a and b — a is the square root of the first term, b is the square root of the second term.
- Write (a + b)(a - b) — plug in your values and simplify.
Common Mistakes
- Using this on addition. x² + 9 is NOT (x + 3)(x + 3). That equals x² + 6x + 9, which is wrong. The pattern requires subtraction.
- Forgetting to check for perfect squares. x² - 7 doesn't factor using this method. 7 isn't a perfect square.
- Swapping the signs in the answer. The binomial with a plus sign comes first. Keep it consistent.
- Not factoring completely. Sometimes your answer can be factored further. Always check.
Difference of Squares vs. Other Patterns
This isn't the only factoring pattern. Here's how to tell them apart:
| Pattern | Formula | When to Use |
|---|---|---|
| Difference of Squares | a² - b² = (a+b)(a-b) | Two squares, subtraction between them |
| Perfect Square Trinomial | a² + 2ab + b² = (a+b)² | Three terms, first and last are squares, middle is twice the product |
| Sum of Cubes | a³ + b³ = (a+b)(a²-ab+b²) | Two cubes, addition between them |
| Difference of Cubes | a³ - b³ = (a-b)(a²+ab+b²) | Two cubes, subtraction between them |
Students mix these up constantly. The difference of squares is the easiest one. Memorize it first.
Practice Problems
Factor each expression:
- x² - 16
- 25m² - 36
- 4p² - 81q²
- 1 - 144k²
- x⁴ - 16
Answers: (x+4)(x-4), (5m+6)(5m-6), (2p+9q)(2p-9q), (1+12k)(1-12k), (x²+4)(x²-4), and x²-4 factors further to (x+2)(x-2), so the complete answer is (x²+4)(x+2)(x-2).
When You'll Use This
Beyond homework, this pattern appears in:
- Rational expressions — simplifying fractions with quadratics
- Solving quadratic equations
- Calculus — u-substitution problems
- Completing the square problems
It shows up everywhere. There's no avoiding it.
The Bottom Line
The difference of squares formula is a tool. Memorize it, apply it when the conditions match, and move on. It takes practice to recognize the pattern instantly, but once you do, these problems become free points.