Difference of Squares Calculator
Factor expressions in the form a² - b² using the formula (a + b)(a - b). Get instant step-by-step solutions.
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Enter values for: (coefficient × value)x² - (coefficient × value)
Understanding Difference of Squares
The difference of squares is one of the most recognizable patterns in algebra. It's a special factoring formula that applies when you have two perfect squares separated by subtraction.
The Formula:
a² - b² = (a + b)(a - b)
Common Examples:
x² - 9 = (x + 3)(x - 3)
4x² - 25 = (2x + 5)(2x - 5)
x² - 16 = (x + 4)(x - 4)
9x² - 1 = (3x + 1)(3x - 1)
x² - 49 = (x + 7)(x - 7)
Recognition Checklist:
- Two terms only (binomial)
- Both terms are perfect squares
- Terms are separated by subtraction (minus sign)
- No middle term (unlike trinomials)
Why It Works:
The formula works because when you multiply (a + b)(a - b) using FOIL, the middle terms cancel:
(a + b)(a - b)
= a² - ab + ab - b²
= a² - b²
Common Mistakes to Avoid:
- Trying to factor the sum of squares (a² + b² does NOT factor over real numbers)
- Forgetting that both terms must be perfect squares
- Missing the subtraction sign between terms
- Not simplifying square roots completely
- Stopping before factoring completely (some factors may factor further)
Frequently Asked Questions
What is the difference of squares formula?
The difference of squares formula is a² - b² = (a + b)(a - b). It's used to factor binomials where two perfect squares are separated by subtraction.
Can you factor the sum of squares?
No, the sum of squares (a² + b²) cannot be factored using real numbers. Only the difference of squares can be factored using this method. This is a common misconception among students.
How do I identify a difference of squares?
Look for three characteristics: (1) exactly two terms, (2) both terms are perfect squares, and (3) the terms are separated by subtraction. If all three conditions are met, you can use the difference of squares formula.
What are perfect squares?
Perfect squares are numbers or expressions that result from squaring whole numbers or simple expressions. Examples include 1, 4, 9, 16, 25, x², 4x², 9x², etc. Both numerical coefficients and variables can be perfect squares.
Can difference of squares be used multiple times?
Yes! Sometimes after factoring once, one or both factors may themselves be differences of squares. For example, x⁴ - 16 = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2). Always check if you can factor further.
How do I verify my factoring is correct?
Multiply your factors back together using FOIL or the distributive property. If you get the original expression, your factoring is correct. The middle terms should cancel out, leaving you with just the two squared terms.
What if the coefficient isn't a perfect square?
First, try factoring out a GCF. For example, 2x² - 8 = 2(x² - 4) = 2(x + 2)(x - 2). Always look for common factors before applying special factoring formulas.
Does order matter in the factors?
No, due to the commutative property of multiplication, (a + b)(a - b) is the same as (a - b)(a + b). Both are correct answers. Some teachers may prefer a specific order, so check your class conventions.
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