Difference of Squares- Definition and Examples
What Is the Difference of Squares?
The difference of squares is a pattern in algebra where you subtract one squared term from another. It looks like this:
a² − b²
That's it. Two perfect squares with a minus sign between them. It doesn't matter what a and b are—they can be numbers, variables, or expressions.
The pattern has a specific factorization that makes solving problems much faster:
a² − b² = (a + b)(a − b)
Why This Works
Multiplying (a + b)(a − b) gives you:
- (a × a) = a²
- (a × −b) = −ab
- (b × a) = ab
- (b × −b) = −b²
The −ab and +ab cancel out. You're left with a² − b².
This isn't magic. It's just basic FOIL multiplication working in reverse. Once you see the pattern, you can't unsee it.
Examples That Make It Clear
Example 1: Numbers
x² − 9
Notice that 9 = 3². So you have x² − 3².
Apply the formula: (x + 3)(x − 3). Done.
Example 2: Variables
4y² − 25
4y² = (2y)² and 25 = 5²
Factor: (2y + 5)(2y − 5)
Example 3: Messier Expressions
(x + 3)² − 16
First, recognize that 16 = 4². So this is (x + 3)² − 4².
Your "a" is (x + 3) and your "b" is 4.
Factor: [(x + 3) + 4][(x + 3) − 4]
Simplify: (x + 7)(x − 1)
How to Spot a Difference of Squares
Not every binomial with a minus sign is a difference of squares. Here's what you need:
- Two terms only
- A minus sign between them
- Both terms must be perfect squares
If any of those three conditions fail, you don't have a difference of squares.
4x² − 8 ← Not a difference of squares. 8 is not a perfect square.
9x² + 16 ← Not a difference of squares. That's addition, not subtraction.
x² − 5y⁴ ← This IS a difference of squares. x² is a perfect square, and 5y⁴ = (y²√5)², so it counts.
Common Factoring Mistakes to Avoid
People get this wrong in predictable ways:
- Forgetting that both terms must be squares
- Factoring x² − 4 as x(x − 4) instead of (x + 2)(x − 2)
- Confusing it with sum of squares, which doesn't factor over real numbers
Difference of Squares vs. Sum of Squares
You need to know the difference. A sum of squares looks the same but has a plus sign:
a² + b²
This does NOT factor into (a + b)(a + b). That's a common misconception. Over real numbers, a² + b² is prime. It only factors if you allow imaginary numbers.
| Type | Form | Factors Over Reals? |
|---|---|---|
| Difference of Squares | a² − b² | Yes: (a + b)(a − b) |
| Sum of Squares | a² + b² | No (unless complex numbers) |
| Perfect Square Trinomial | (a ± b)² | Already factored |
Getting Started: How to Factor Difference of Squares
Follow these steps in order:
- Check the sign. You need a minus sign, not a plus.
- Verify both terms are squares. If one isn't, stop here.
- Find the square roots. √a² = a, √b² = b
- Write the factors. (a + b)(a − b)
- Check by multiplying. (a + b)(a − b) should give you a² − b²
That's the entire process. Practice with ten problems and you'll have it locked in.
Where This Shows Up
Difference of squares shows up in:
- Factoring polynomials on tests
- Simplifying rational expressions
- Solving quadratic equations
- Rationalizing denominators
It's one of those foundational skills. You learn it in algebra, but it keeps appearing in precalculus, calculus, and beyond. Master it now or struggle later.
Quick Practice
Factor these (answers at bottom):
- 16 − x²
- 49y² − 121
- (2x − 1)² − 36
Answers:
- (4 + x)(4 − x) or (x + 4)(x − 4) with sign flipped
- (7y + 11)(7y − 11)
- (2x − 1 + 6)(2x − 1 − 6) = (2x + 5)(2x − 7)
If you got those without help, you understand the pattern. If not, go back and check which step you missed.