MATH LESSON
PRODUCT OF SUM & DIFFERENCE
When multiplying a sum and difference of the same two terms, the middle terms cancel!
When multiplying a sum and difference of the same two terms, the middle terms cancel, leaving a difference of squares.
(a + b)(a − b) = a² − b²
(a + b)
×
(a − b)
=
a² − b²
Understanding the Pattern
When you multiply (a + b) by (a − b), the result has only 2 terms — NOT 3! The middle terms cancel each other out because they are opposites.
1st term: a² — first term squared
2nd term: − b² — second term squared, always negative
NO middle term! It cancelled out!
Why Do the Middle Terms Cancel?
Use FOIL on (a + b)(a − b) and watch what happens:
(a + b)(a − b)
F: a · a = a²
O: a · (−b) = −ab
I: b · a = +ab
L: b · (−b) = −b²
O + I: −ab + +ab = 0 (CANCELED!)
Final: a² − b²
Order Doesn't Matter!
Whether you write the sum first or the difference first, the answer is the same:
(a + b)(a − b) = a² − b²
(a − b)(a + b) = a² − b²
Both give the EXACT same result!
Common Mistakes!
Mistake 1: Writing a² + b² instead of a² − b²
(x+5)(x−5) = x² + 25 × WRONG
(x+5)(x−5) = x² − 25 ✓ CORRECT
Mistake 2: Forgetting to square each term.
(x+5)(x−5) = x − 5 × WRONG
(x+5)(x−5) = x² − 25 ✓ CORRECT
Worked Examples
Example 1: (x + 5)(x − 5)
a = x → x² | b = 5 → 5² = 25
→ x² − 25
a = x → x² | b = 5 → 5² = 25
→ x² − 25
Example 2: (2x + 3)(2x − 3)
a = 2x → (2x)² = 4x² | b = 3 → 3² = 9
→ 4x² − 9
a = 2x → (2x)² = 4x² | b = 3 → 3² = 9
→ 4x² − 9
Example 3: (3x − 4)(3x + 4)
a = 3x → (3x)² = 9x² | b = 4 → 4² = 16
→ 9x² − 16 (order swapped, same answer!)
a = 3x → (3x)² = 9x² | b = 4 → 4² = 16
→ 9x² − 16 (order swapped, same answer!)
Example 4: (x + 10)(x − 10)
a = x → x² | b = 10 → 10² = 100
→ x² − 100
a = x → x² | b = 10 → 10² = 100
→ x² − 100
Example 5: (5x + 2)(5x − 2)
a = 5x → (5x)² = 25x² | b = 2 → 2² = 4
→ 25x² − 4
a = 5x → (5x)² = 25x² | b = 2 → 2² = 4
→ 25x² − 4
Example 6: (4x − 5)(4x + 5)
a = 4x → (4x)² = 16x² | b = 5 → 5² = 25
→ 16x² − 25
a = 4x → (4x)² = 16x² | b = 5 → 5² = 25
→ 16x² − 25
Example 7: (x + 12)(x − 12)
a = x → x² | b = 12 → 12² = 144
→ x² − 144
a = x → x² | b = 12 → 12² = 144
→ x² − 144
Example 8: (8x + 3)(8x − 3)
a = 8x → (8x)² = 64x² | b = 3 → 3² = 9
→ 64x² − 9
a = 8x → (8x)² = 64x² | b = 3 → 3² = 9
→ 64x² − 9
Shortcut: Just square the first term, then square the second term, and put a minus between them. No middle term needed!
MATH SEATWORK
Seatwork Time!
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