Difference of Squares Calculator
Result:
Master the Difference of Squares factoring technique with our comprehensive calculator. Learn to factor expressions of the form a² - b² using the fundamental algebraic identity.
What is the Difference of Squares?
The difference of squares is a special factoring pattern where a² - b² = (a + b)(a - b). This identity is one of the most important factoring formulas in algebra.
The Difference of Squares Formula
a² - b² = (a + b)(a - b)
Recognition Pattern
Must have exactly 2 terms
Connected by subtraction (-)
Both terms are perfect squares
Examples:
- x² - 9 = x² - 3²
- 25 - y² = 5² - y²
- 16x² - 49 = (4x)² - 7²
Factoring Steps
Step 1: Identify a² and b²
Step 2: Find square roots a and b
Step 3: Write as (a + b)(a - b)
Example: x² - 16
- a = x, b = 4
- (x + 4)(x - 4)
Detailed Examples
Simple Variables
Factor: x² - 25
Solution:
a = x, b = 5
Answer: (x + 5)(x - 5)
With Coefficients
Factor: 4x² - 9
Solution:
a = 2x, b = 3
Answer: (2x + 3)(2x - 3)
Large Numbers
Factor: 100 - y²
Solution:
a = 10, b = y
Answer: (10 + y)(10 - y)
Complex Difference of Squares
Multi-Step Example: Factor 81x⁴ - 16y²
Step 1: Recognize as difference of squares
81x⁴ - 16y² = (9x²)² - (4y)²
Step 2: Apply the formula
= (9x² + 4y)(9x² - 4y)
Step 3: Check if further factoring is possible
9x² + 4y cannot be factored further (sum of squares)
9x² - 4y cannot be factored further (not perfect squares)
Final Answer: (9x² + 4y)(9x² - 4y)
Common Mistakes to Avoid
❌ Wrong Approaches
Sum of squares: a² + b² ≠ (a + b)²
Sign error: Forgetting the negative sign
Not perfect squares: Trying to factor non-squares
Example: x² + 9 cannot be factored using real numbers
✅ Correct Approaches
Check pattern: Exactly 2 terms with subtraction
Verify squares: Confirm both terms are perfect squares
Apply formula: a² - b² = (a + b)(a - b)
Double-check: Multiply back to verify
Applications in Problem Solving
- Solving equations: Set each factor equal to zero
- Simplifying fractions: Cancel common factors
- Graphing: Find x-intercepts of quadratic functions
- Geometry: Area and perimeter calculations
- Physics: Energy and motion equations
Solving Equations Using Difference of Squares
Example: Solve x² - 36 = 0
Step 1: Factor using difference of squares
x² - 36 = (x + 6)(x - 6) = 0
Step 2: Apply zero product property
Either x + 6 = 0 or x - 6 = 0
Step 3: Solve each equation
x + 6 = 0 → x = -6
x - 6 = 0 → x = 6
Solutions: x = -6 and x = 6
Special Cases
Nested Squares
(x²)² - 9 = x⁴ - 9
= (x² + 3)(x² - 3)
Fourth powers as squaresFractional Terms
x² - 1/4 = x² - (1/2)²
= (x + 1/2)(x - 1/2)
Fractions as perfect squaresMultiple Variables
x²y² - z² = (xy)² - z²
= (xy + z)(xy - z)
Products as single termsPractice Problems
Try factoring these expressions:
- x² - 64 → Answer: (x + 8)(x - 8)
- 25y² - 1 → Answer: (5y + 1)(5y - 1)
- 144 - 4x² → Answer: 4(6 + x)(6 - x)
- a⁴ - 81 → Answer: (a² + 9)(a² - 9) = (a² + 9)(a + 3)(a - 3)
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