الأسبوع 6 · الحالات الخاصة: المربع الكامل والفرق بين مربعين ومجموع مكعبين والفرق بين مكعبين · 3 دروس
2
Difference of two squares and quick mental arithmetic
Mathematics Prep 2 Term 1, Unit 2, Lesson 3
GoalFactorise a² − b² as (a − b)(a + b), also after taking out a common factor and repeatedly, and use it for quick sums.
New words
difference of two squares · a² − b², two perfect squares joined by a minus signconjugate brackets · (a − b) and (a + b)sum of squares · a² + b² cannot be factorised over the real numbersfactorise completely · keep going until no bracket can be factorised further
Explanation
(a − b)(a + b) = a² + ab − ab − b² = a² − b². So a² − b² = (a − b)(a + b). With coefficients: 4x² − 25y² = (2x)² − (5y)² = (2x − 5y)(2x + 5y). Take out a common factor first: 3x² − 75 = 3(x² − 25) = 3(x − 5)(x + 5). Sometimes you can factorise twice: x⁴ − 16 = (x² − 4)(x² + 4) = (x − 2)(x + 2)(x² + 4). Mental arithmetic gets easy: 99² − 1 = (99 − 1)(99 + 1) = 98 × 100 = 9800, and 1002 × 998 = 1000² − 2² = 999 996. If x − y = 3 and x² − y² = 21, then (x − y)(x + y) = 21, so x + y = 7. Careful: x² + 9 and x² + 25 are sums, not differences, and cannot be factorised over the real numbers.
Expression
Factorised
x² − 49
(x − 7)(x + 7)
4x² − 25y²
(2x − 5y)(2x + 5y)
3x² − 75
3(x − 5)(x + 5)
x⁴ − 16
(x − 2)(x + 2)(x² + 4)
Worked examples
Factorise 9x² − 16y². 9x² = (3x)² and 16y² = (4y)², so the answer is (3x − 4y)(3x + 4y). Check: 9x² + 12xy − 12xy − 16y².
A square lawn has side 12 m, and a square corner of side 5 m is cut off for a path. The remaining area is 12² − 5² = (12 − 5)(12 + 5) = 7 × 17 = 119 m². Check: 144 − 25 = 119.
Factorise x⁴ − 81 completely. x⁴ − 81 = (x² − 9)(x² + 9) = (x − 3)(x + 3)(x² + 9). The bracket x² + 9 is a sum of squares, so we stop. Also find 101² − 99² = (101 − 99)(101 + 99) = 2 × 200 = 400.
Class activity
Quick squares. Use a² − b² = (a − b)(a + b) to find 52 × 48 (write it as 50² − 2²), 31 × 29 and 1005 × 995 without a calculator. Check one answer by long multiplication.
Practice
1
Factorise x² − 49.
(x − 7)(x + 7).
2
Factorise 2x² − 50.
2(x² − 25) = 2(x − 5)(x + 5).
3
Find 52 × 48 using 50² − 2².
2500 − 4 = 2496.
4
If x − y = 2 and x² − y² = 14, find x + y.
(x − y)(x + y) = 14, so x + y = 7.
5
Seif says x² + 16 = (x − 4)(x + 4). Is he right?
No. (x − 4)(x + 4) = x² − 16. A sum of two squares does not factorise over the real numbers.
