Difference of Squares Calculator
Factor expressions like x² − 49, 25a² − 16b², and y⁴ − 81 instantly using the Difference of Squares Formula. Get accurate answers, complete factorization, and step-by-step explanations in seconds.
What is a Difference of Squares?
A Difference of Squares occurs when two perfect square terms are separated by a minus sign. This special algebraic pattern can be factored instantly using one of the most important identities in algebra.
To apply the Difference of Squares Formula correctly, the expression must satisfy all three conditions below.
Perfect Square
a² must be a perfect square.
Minus Sign
The two terms must be separated by subtraction.
Perfect Square
b² must also be a perfect square.
Quick Example
Difference of Squares factorization appears frequently in algebra, polynomial simplification, quadratic expressions, and equation solving. Our calculator automatically identifies the pattern and applies the correct formula instantly.
How to Identify a Difference of Squares
Before applying the Difference of Squares Formula, make sure the expression follows the correct pattern. Use these four simple checks.
Check the First Term
The first term must be a perfect square such as x², 25a², 49m², or 81y⁴.
Look for a Minus Sign
A Difference of Squares always uses subtraction. Expressions with addition do not qualify.
Check the Second Term
The second term must also be a perfect square such as 9, 16b², 64, or n².
Apply the Formula
Once both terms are perfect squares, use the identity: a² − b² = (a + b)(a − b)
Example
How to Factor Using the Difference of Squares Formula
Once you identify a Difference of Squares expression, the factorization process becomes simple. Follow these four steps to factor any valid expression.
Identify Perfect Squares
Check whether both terms are perfect squares. Examples include x², 25, 49a², and 81b⁴.
Rewrite Each Square
Express both terms as squares of simpler expressions.
Apply the Formula
Use the identity a² − b² = (a + b)(a − b).
Verify the Answer
Multiply the factors together to confirm that you get the original expression.
Worked Example
= (7x)² − 4²
= (7x + 4)(7x − 4)
Difference of Squares Examples
Below are some common examples of the Difference of Squares formula. Notice how each expression follows the pattern a² − b².
x² − 9
(x + 3)(x − 3)
25x² − 49
(5x + 7)(5x − 7)
m² − 64n²
(m + 8n)(m − 8n)
x⁴ − 16
(x² + 4)(x + 2)(x − 2)
Factor 25x² − 49
25x² = (5x)²
49 = 7²
a² − b² = (a + b)(a − b)
25x² − 49 = (5x + 7)(5x − 7)
When Difference of Squares Doesn’t Work
The Difference of Squares formula only works when you have exactly two perfect squares separated by a minus sign. Here are the most common situations where the formula cannot be applied.
Addition Instead of Subtraction
This expression is a sum of squares, not a difference of squares. The formula only works when the terms are separated by a minus sign.
Not Perfect Squares
The number 5 is not a perfect square, so the expression does not match the pattern a² − b².
More Than Two Terms
This is a trinomial, not a two-term expression. Different factoring techniques are required.
Quick Rule to Remember
Before applying the formula, ask yourself:
- Are there exactly two terms?
- Is there a minus sign between them?
- Are both terms perfect squares?
If the answer is yes to all three questions, you can safely use:
Frequently Asked Questions
What happens if the power is higher than 2?
x⁴ − 16 = (x² − 4)(x² + 4)
However, if the exponent is odd (such as x³), a different identity may apply.
Why doesn’t a² + b² have a similar factoring formula?
Is 1 always a perfect square?
Can I use the Difference of Squares formula for numbers only?
Difference of Squares: Key Takeaway
The Difference of Squares is one of the most useful algebraic identities for factoring expressions that contain two perfect squares separated by a minus sign.
Instead of factoring manually every time, use our calculator to instantly identify and factor Difference of Squares expressions with step-by-step explanations.
Try the Difference of Squares Calculator