Difference of Squares Calculator
Factor expressions in the form a² - b² using the difference of squares formula. This calculator shows the step-by-step factorization process and verifies the result.
Formula:
a² - b² = (a + b)(a - b)
Example: 5² - 3² = (5 + 3)(5 - 3) = 8 × 2 = 16
• Works for any real numbers a and b
• Useful for factoring quadratic expressions
• Shows step-by-step factorization process
• Verifies results with expanded form
What is the Difference of Squares?
The difference of squares is a special algebraic pattern where one squared term is subtracted from another. It can always be factored using the formula: a² - b² = (a + b)(a - b).
Formula: a² - b² = (a + b)(a - b)
Example: 16 - 4 = 4² - 2²
= (4 + 2)(4 - 2)
= (6)(2) = 12
Example Problems
Basic Examples
Simple Numbers
- 9 - 4 = 3² - 2² = (3 + 2)(3 - 2) = 5 × 1 = 5
- 25 - 16 = 5² - 4² = (5 + 4)(5 - 4) = 9 × 1 = 9
- 100 - 36 = 10² - 6² = (10 + 6)(10 - 6) = 16 × 4 = 64
With Variables
- x² - 1 = (x + 1)(x - 1)
- x² - 4 = (x + 2)(x - 2)
- x² - 9 = (x + 3)(x - 3)
Real-World Applications
Area Problems
Find the difference in area between two squares:
Large square: 8 × 8 = 64
Small square: 5 × 5 = 25
Difference in area:
64 - 25 = 8² - 5² = (8 + 5)(8 - 5) = 13 × 3 = 39
Number Theory
Use difference of squares to factor numbers:
Factor 399:
Solution:
399 = 400 - 1
= 20² - 1²
= (20 + 1)(20 - 1)
= 21 × 19
Key Points to Remember
- The formula only works for expressions in the form a² - b²
- The terms must be perfect squares
- The middle term must be subtraction (-)
- The result is always two binomial factors
- This pattern is one of the standard factoring techniques