الأسبوع 6 · الحالات الخاصة: المربع الكامل والفرق بين مربعين ومجموع مكعبين والفرق بين مكعبين · 3 دروس
1
Perfect-square trinomials
Mathematics Prep 2 Term 1, Unit 2, Lesson 3
GoalRecognise a perfect-square trinomial, factorise it as (a ± b)², and find a missing term or coefficient.
New words
perfect square · an expression that is the square of another, such as (x + 3)²perfect-square trinomial · a² + 2ab + b² or a² − 2ab + b²middle term test · the middle term is twice the product of the roots of the outer termsmissing term · the term that makes the trinomial a perfect square
Explanation
Expanding (a + b)² gives a² + 2ab + b², and (a − b)² gives a² − 2ab + b². Read backwards, these are the perfect-square trinomials: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². The picture shows why: a square of side x + 3 is cut into x², two rectangles 3x and a small square 9. To test a trinomial: the first and last terms must be perfect squares with plus signs, and the middle term must be twice the product of their roots. For 4x² − 12x + 9: the roots are 2x and 3, and 2 × 2x × 3 = 12x, so it equals (2x − 3)². Be careful: (a + b)² is not a² + b² (for a = 3 and b = 4, 49 is not 25), and x² + 25 is not a perfect square.
Worked examples
Factorise x² + 10x + 25. First term x² = (x)², last term 25 = 5², and the middle term 10x = 2 × x × 5. So x² + 10x + 25 = (x + 5)².
Find the missing term. x² + 14x + a: the roots are x and √a, and 2 × x × √a = 14x, so √a = 7 and a = 49. For x² − ax + 64: √64 = 8 and 2 × 8 = 16, so −a = ±16 and a = 16 or a = −16. Both work: x² − 16x + 64 = (x − 8)² and x² + 16x + 64 = (x + 8)².
Mental arithmetic: 57² + 2 × 57 × 43 + 43² = (57 + 43)² = 100² = 10 000. Check the shape of the expression first: a² + 2ab + b² with a = 57 and b = 43.
Class activity
Square tiles. Cut four pieces of card: a square x by x, two rectangles x by 2, and a square 2 by 2 (choose x = 5 cm). Put them together into one big square. What is its side? What does this show about x² + 4x + 4?
Practice
1
Factorise x² + 6x + 9.
(x + 3)².
2
Factorise 9x² − 30x + 25.
3x and 5 are the roots, and 2 × 3x × 5 = 30x, so (3x − 5)².
3
Is x² + 8x + 9 a perfect square? Explain.
No. The roots are x and 3, and 2 × x × 3 = 6x, not 8x.
4
Find 98² + 2 × 98 × 2 + 2² quickly.
(98 + 2)² = 100² = 10 000.
5
Mostafa writes (x + 4)² = x² + 16. What is the right answer?
x² + 2 × x × 4 + 16 = x² + 8x + 16. He forgot the middle term 8x.
