Factorization: a² − b² and a² ± 2ab + b²
Recognizing Special Patterns
Essential Understanding: Factorization is the reverse of expanding brackets. Some expressions follow special patterns that allow us to factor them quickly without trial and error. Mastering these patterns is essential for solving equations and simplifying algebraic fractions.
Difference of Squares: \(a^2 - b^2\)
Perfect Square (+): \(a^2 + 2ab + b^2\)
Perfect Square (−): \(a^2 - 2ab + b^2\)
Pattern 1: Difference of Two Squares
The Formula
\[ a^2 - b^2 = (a + b)(a - b) \]
In words: The difference of two squares equals the sum times the difference.
\(a^2 - b^2\)
→
\((a + b)(a - b)\)
How to Recognize It
- Two terms separated by a minus sign
- Both terms are perfect squares
- Examples of perfect squares: \(1, 4, 9, 16, 25, x^2, 4y^2, 9z^4\)
Example 1: Factorize \(x^2 - 9\)
1
Identify the squares:
\(x^2 = (x)^2\) and \(9 = (3)^2\)
\(x^2 = (x)^2\) and \(9 = (3)^2\)
2
Apply the formula:
\(a = x\), \(b = 3\)
\(x^2 - 9 = (x + 3)(x - 3)\)
\(a = x\), \(b = 3\)
\(x^2 - 9 = (x + 3)(x - 3)\)
Example 2: Factorize \(4x^2 - 25y^2\)
1
Identify the squares:
\(4x^2 = (2x)^2\) and \(25y^2 = (5y)^2\)
\(4x^2 = (2x)^2\) and \(25y^2 = (5y)^2\)
2
Apply the formula:
\(a = 2x\), \(b = 5y\)
\(4x^2 - 25y^2 = (2x + 5y)(2x - 5y)\)
\(a = 2x\), \(b = 5y\)
\(4x^2 - 25y^2 = (2x + 5y)(2x - 5y)\)
Pattern 2: Perfect Square Trinomials
Addition Pattern
\[ a^2 + 2ab + b^2 = (a + b)^2 \]
In words: First squared, plus twice the product, plus second squared equals the sum squared.
Subtraction Pattern
\[ a^2 - 2ab + b^2 = (a - b)^2 \]
In words: First squared, minus twice the product, plus second squared equals the difference squared.
How to Recognize Perfect Square Trinomials
- Three terms (trinomial)
- First and last terms are perfect squares
- Middle term = 2 × (square root of first) × (square root of last)
Test: For \(x^2 + 10x + 25\): Is \(10 = 2 \times x \times 5\)? Yes! So it's \((x+5)^2\)
Example 3: Factorize \(x^2 + 6x + 9\)
1
Check first and last terms:
\(x^2 = (x)^2\) ✓ and \(9 = (3)^2\) ✓
\(x^2 = (x)^2\) ✓ and \(9 = (3)^2\) ✓
2
Check middle term:
Is \(6x = 2 \times x \times 3\)? Yes! \(6x = 6x\) ✓
Is \(6x = 2 \times x \times 3\)? Yes! \(6x = 6x\) ✓
3
Factor (middle term is positive, so use +):
\(x^2 + 6x + 9 = (x + 3)^2\)
\(x^2 + 6x + 9 = (x + 3)^2\)
Example 4: Factorize \(4x^2 - 12x + 9\)
1
Check first and last terms:
\(4x^2 = (2x)^2\) ✓ and \(9 = (3)^2\) ✓
\(4x^2 = (2x)^2\) ✓ and \(9 = (3)^2\) ✓
2
Check middle term:
Is \(12x = 2 \times 2x \times 3\)? Yes! \(12x = 12x\) ✓
Is \(12x = 2 \times 2x \times 3\)? Yes! \(12x = 12x\) ✓
3
Factor (middle term is negative, so use −):
\(4x^2 - 12x + 9 = (2x - 3)^2\)
\(4x^2 - 12x + 9 = (2x - 3)^2\)
Interactive Factorizer
Difference of Squares Calculator
Enter values for \(a\) and \(b\) to see the factorization of \(a^2 - b^2\):
\(x^2 - 25 = (x + 5)(x - 5)\)
Practice Problems
Factorize: \(x^2 - 16\)
\((x + 4)(x - 4)\)
Factorize: \(9a^2 - 1\)
\((3a + 1)(3a - 1)\)
Factorize: \(x^2 + 8x + 16\)
\((x + 4)^2\)
Factorize: \(y^2 - 14y + 49\)
\((y - 7)^2\)
Factorize: \(16m^2 - 49n^2\)
\((4m + 7n)(4m - 7n)\)
Factorize: \(25x^2 + 30x + 9\)
\((5x + 3)^2\)
CSEC Examination Questions
Test Your Understanding
1
Factorize completely: \(x^2 - 49\)
Solution: \(x^2 - 49 = x^2 - 7^2 = (x + 7)(x - 7)\)
This is a difference of two squares.
This is a difference of two squares.
2
Which expression equals \((3x - 2)^2\)?
Solution: \((3x - 2)^2 = (3x)^2 - 2(3x)(2) + (2)^2 = 9x^2 - 12x + 4\)
3
Factorize: \(100 - 81y^2\)
Solution: \(100 - 81y^2 = 10^2 - (9y)^2 = (10 + 9y)(10 - 9y)\)
4
Factorize completely: \(2x^2 - 50\)
Solution: First factor out the common factor 2:
\(2x^2 - 50 = 2(x^2 - 25) = 2(x + 5)(x - 5)\)
Always look for common factors first!
\(2x^2 - 50 = 2(x^2 - 25) = 2(x + 5)(x - 5)\)
Always look for common factors first!
Key Formulas to Memorize
| Difference of Squares: | \(a^2 - b^2 = (a + b)(a - b)\) |
| Perfect Square (+): | \(a^2 + 2ab + b^2 = (a + b)^2\) |
| Perfect Square (−): | \(a^2 - 2ab + b^2 = (a - b)^2\) |
CSEC Examination Tips
- Always check for common factors first before applying special patterns
- \(a^2 + b^2\) cannot be factorized — the sum of squares has no factors
- Verify your answer by expanding the brackets to check
- For perfect squares, check the middle term equals \(2ab\)
- Remember: \((a-b)^2 \neq a^2 - b^2\) — these are different!