Mastering Quadratics: Multiplication, Factorization & Expansion Techniques

1. Stage 8 Chapter 18 Quadratics

2. Objectives • Multiply expressions of the form (x+3)(x-7) and simplify the resulting expression; • factorise quadratic expressions including the difference of two squares

3. You should already know • How to collect together simple algebraic terms • Expand single brackets • Take out common factors

4. Expanding two brackets Look at this algebraic expression: (3 + t)(4 – 2t) This means (3 + t)× (4 – 2t), but we do not usually write × in algebra. To expand or multiply out this expression we multiply every term in the second bracket by every term in the first bracket. (3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t) This is a quadratic expression. = 12 – 6t + 4t – 2t2 = 12 – 2t – 2t2

5. Complete the activityUsing the grid method to expand brackets

6. Expanding two brackets With practice we can expand the product of two linear expressions in fewer steps. For example, – 10 (x – 5)(x + 2) = + 2x – 5x x2 = x2 – 3x – 10 Notice that –3 is the sum of –5 and 2 … … and that –10 is the product of –5 and 2.

7. Complete the activityMatching quadratic expressions 1

8. Complete the activityMatching quadratic expressions 2

9. Squaring expressions Expand and simplify: (2 – 3a)2 We can write this as, (2 – 3a)2 = (2 – 3a)(2 – 3a) Expanding, 2(2 – 3a) – 3a(2 – 3a) (2 – 3a)(2 – 3a) = = 4 – 6a – 6a + 9a2 = 4 – 12a + 9a2

10. Squaring expressions In general, (a + b)2 = a2 + 2ab + b2 The first term squared … … plus 2 × the product of the two terms … … plus the second term squared. For example, (3m + 2n)2 = 9m2 + 12mn + 4n2

11. Complete the activitySquaring expressions

12. The difference between two squares Expand and simplify (2a + 7)(2a – 7) Expanding, 2a(2a – 7) + 7(2a – 7) (2a + 7)(2a – 7) = – 49 = – 14a + 14a 4a2 = 4a2 – 49 When we simplify, the two middle terms cancel out. This is the difference between two squares. In general, (a + b)(a – b)= a2 – b2

13. Complete the activityThe difference between two squares

14. Complete the activityMatching the difference between two squares

15. Quadratic expressions t2 ax2 + bx + c (where a = 0) 2 A quadratic expression is an expression in which the highest power of the variable is 2. For example, x2 – 2, w2 + 3w + 1, 4 – 5g2 , The general form of a quadratic expression in x is: x is a variable. a is a fixed number and is the coefficient of x2. b is a fixed number and is the coefficient of x. c is a fixed number and is a constant term.

16. Expanding or multiplying out a2 + 3a + 2 (a + 1)(a + 2) Factorizing Remember: factorizing an expression is the opposite of expanding it. Factorizing expressions Often: When we expand an expression we remove the brackets. When we factorize an expression we write it with brackets.

17. Factorizing quadratic expressions Factorise x² +7x + 12 This will factorise into two brackets with x as the first term in each x² +7x + 12 = (x )(x ) As both the signs are positive, both the numbers will be positive You need to find two numbers that multiply together to give 12 and add together to give 7 These will be +3 and +4 So x² +7x + 12 = (x + 3)(x + 4) or x² +7x + 12 = (x + 4)(x + 3)

18. Complete the activityFactorizing quadratic expressions 1

19. Complete the activityMatching quadratic expressions 1

20. Factorizing quadratic expressions Quadratic expressions of the form ax2 + bx + c can be factorized if they can be written using brackets as (dx + e)(fx + g) where d, e, f and g are integers. If we expand (dx + e)(fx + g)we have, (dx + e)(fx + g)= dfx2 + dgx + efx + eg = dfx2 + (dg + ef)x + eg Comparing this to ax2 + bx + cwe can see that we must choose d, e, f and g such that: a = df, b = (dg + ef) c = eg

21. Complete the activityFactorizing quadratic expressions 2

22. Matching quadratic expressions 2

23. Factorizing the difference between two squares x2 – a2 = (x + a)(x – a) A quadratic expression in the form x2 – a2 is called the difference between two squares. The difference between two squares can be factorized as follows: For example, 9x2 – 16= (3x + 4)(3x – 4) 25a2 – 1= (5a + 1)(5a – 1) m4 – 49n2 = (m2 + 7n)(m2 – 7n)

24. Factorizing the difference between two squares

25. Matching the difference between two squares

26. Key idas • When multiplying two brackets, multiply every term in the first bracket by every term in the second • To factorise x²+ax+b: if b is positive find two numbers that multiply to give b and add up to a • To factorise x²+ax+b: if b is negative find two numbers that multiply to give b and have a difference of a • The difference of two squares factorises • x²- a² = (x+a)(x-a)