Step by Step
1
Recognize the pattern
Look for two perfect squares being subtracted — an expression in the form a² − b².
2
Identify a and b
Take the square root of each term to find a and b. For x² − 9, a = x and b = 3 (since 9 = 3²).
3
Write the factored form
a² − b² always factors into (a + b)(a − b) — one factor with addition, one with subtraction.
4
Know when NOT to use it
This pattern only works with subtraction. A sum of squares (a² + b²) does not factor this way over real numbers.
Applied Walkthrough
1
Factor x² − 9. Recognize this as a difference of two perfect squares: x² and 9 = 3².
2
Identify a = x and b = 3.
3
Apply the pattern: x² − 9 = (x + 3)(x − 3).
4
Contrast this with x² + 25 — since this is addition, not subtraction, it does NOT factor into (x+5)(x+5); that would actually expand to x² + 10x + 25, a completely different expression.
Exam Application
Exams test whether you can recognize the difference-of-squares pattern at a glance, correctly identify a and b, and — critically — know that the pattern only applies to subtraction, not addition.
⚠ Common Trap
The most common trap is applying this pattern to a sum of squares (a² + b²) instead of a difference. A plus sign between two perfect squares does not factor this way at all over real numbers — leave it alone.
✓ Quick Self-Check
1. What is the difference of squares factoring pattern?
a² − b² = (a + b)(a − b).
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2. Factor x² − 9.
(x + 3)(x − 3).
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3. Does x² + 25 factor using this pattern?
No — the pattern only applies to subtraction, not addition, of two perfect squares.
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4. In a² − b² = (a+b)(a−b), what are a and b for x² − 49?
a = x, b = 7.
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5. What would (x+5)(x+5) actually expand to?
x² + 10x + 25 — not x² + 25, which is why the sum-of-squares version doesn't factor this way.
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