1. 11/06/2016 1 Factorizing Quadratic Equations

2. 11/06/2016 2 x 2 ± bx ± c x 2 ± bx ± c If this is positive, both signs are the same. The numbers ADD to give this value and MULTIPLY to give this value. If this is negative, both signs are different. If this is positive, both signs are +. If this is negative, both signs are -. The numbers have a DIFFERENCE of this value The largest value takes this sign.

3. 11/06/2016 3 eg. x 2 + 5x + 4 The numbers ADD to give 5 and MULTIPLY to give 4. In other words 4 and 1. This is positive, so both signs are +. Answer (x+4)(x+1) is positive, so both signs are the same. This is positive, so both signs are the same.

4. 11/06/2016 4 eg. x 2 - 10x + 16 his is positive, so both signs are the same. This is positive, so both signs are the same. The numbers ADD to give 10 and MULTIPLY to give 16.In other words 2 and 8. This is negative, so both signs are -. Answer (x-2)(x-8)

5. 11/06/2016 5 eg. x 2 - 6x - 16 The numbers MULTIPLY to give this value and have a DIFFERENCE of this value. In other words 2 and 8. This is negative, so both signs are different. The largest value takes this sign. Answer (x+2)(x-8)

6. 11/06/2016 6 eg. x 2 + 4x - 32 The numbers MULTIPLY to give this value and have a DIFFERENCE of this value. In other words 4 and 8. This is negative, so both signs are different. The largest value takes this sign. Answer (x-4)(x+8)

7. 11/06/2016 7 Exercise 1. x 2 + 6x + 5 2. x 2 - 6x + 5 3. x 2 + 8x + 16 4. x 2 - 10x + 16 5. x 2 - 4x – 12 6. x 2 + 6x – 16 7. x 2 + 15x – 16 8. x 2 – 7x -30 9. x 2 + x – 20 10. x 2 – 16 1. (x+5)(x+1) 2. (x-5)(x-1) 3. (x+4)(x+4) 4. (x-8)(x-2) 5. (x-6)(x+2) 6. (x+8)(x-2) 7. (x+16)(x-1) 8. (x-10)(x+3) 9. (x+5)(x-4) 10.(x+4)(x-4)

8. 11/06/2016 8 eg. x 2 ± 0x - 16 The numbers MULTIPLY to give this value and have a DIFFERENCE of this value. In other words 4 and 4. This is negative, so both signs are different. The largest value takes this sign. Irrelevant in this case! Answer (x+4)(x-4)

9. 11/06/2016 9 What happens when there is more than one lot of x 2, i.e. the general case of ax 2 ± bx ± c There is a slight change here. There is a slight change here. First of all multiply a and c. First of all multiply a and c. We are now looking for 2 values that multiply to give (a x c) and either add to give, or have a difference of b. We are now looking for 2 values that multiply to give (a x c) and either add to give, or have a difference of b. We must now rewrite the equation and look to factorise the two separate parts of the equation to give a common factor. We must now rewrite the equation and look to factorise the two separate parts of the equation to give a common factor.

10. 11/06/2016 10 eg. 2x 2 + 9x + 4 (2x4 = + 8) The numbers ADD to give 9 and MULTIPLY to give 8. In other words 8 and 1. This is positive, so both signs are +. Rewrite 2x 2 +8x +1x + 4 is positive, so both signs are the same. This is positive, so both signs are the same. Factorize 2x(x + 4) +1(x + 4) Answer (2x+1)(x + 4)

11. 11/06/2016 11 eg. 3x 2 - 5x - 2 (3x - 2= - 6) The numbers HAVE A DIFFERENCE OF 5 and MULTIPLY to give 6. In other words 6 and 1. This is negative, so the largest value is -. Rewrite 3x 2 -6x +1x - 2 is negative, so the signs are different. This is negative, so the signs are different. Factorize 3x(x - 2) +1(x - 2) Answer (3x+1)(x-2)

12. 11/06/2016 12 eg. 8x 2 - 10x - 3 (8 x - 3= - 24) The numbers HAVE A DIFFERENCE OF 10 and MULTIPLY to give 24. In other words 12 and 2. This is negative, so the largest value is -. Rewrite 8x 2 -12x +2x - 3 is negative, so the signs are different. This is negative, so the signs are different. Factorize 4x(2x - 3) +1(2x - 3) Answer (4x+1)(2x-3)

13. 11/06/2016 13 Exercise 1. 2x 2 + 5x + 3 2. 2x 2 + 7x + 3 3. 3x 2 + 7x + 2 4. 2x 2 - x - 15 5. 2x 2 + x – 21 6. 3x 2 - 17x – 28 7. 6x 2 + 7x - 3 8. 10x 2 + 9x + 2 9. 12x 2 + 23x + 10 10. 6x 2 – 27x + 30 1. (2x+3)(x+1) 2. (2x+1)(x+3) 3. (3x+1)(x+2) 4. (2x+5)(x-3) 5. (2x+7)(x-3) 6. (3x+4)(x-7) 7. (2x+3)(3x-1) 8. (5x+2)(2x+1) 9. (3x+2)(4x+5) 10.3(2x-5)(x-2)