Quadratic Equations Practice: Factorization and Solutions

Explore solutions to quadratic equations by factorizing them and finding the roots. Learn how to solve various quadratic equations step by step with examples provided in the practice questions.

• Quadratic Equations
• Factorization
• Solutions
• Practice
• Algebra

arm_biz Follow

Uploaded on Oct 10, 2024 | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

Presentation Transcript

1. CHAPTER 04 Algebra III Quadratic Relations Solutions: Practice Questions 4.3

2. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: 8x2 16x = 0 (i) 8x2 16x = 0 8x (x 2) = 0 8x = 0 and x 2 = 0 x = 0 and x = 2

3. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: 6x2 + 3x = 0 (ii) 6x2 + 3x = 0 3x(2x + 1) = 0 3x = 0 and 2x + 1 = 0 x = 0 2x = 1 x =- 1 2

4. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: 16x2 9 = 0 (iii) 16x2 9 = 0 (4x)2 (3)2 = 0 (4x 3)(4x + 3) = 0 4x 3 = 0 and 4x + 3 = 0 4x = 3 and 4x = 3 x =3 x =-3 and 4 4

5. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: p2 8p + 12 = 0 (iv) p2 8p + 12 = 0 (p 2)(p 6) = 0 p 2 = 0 and p 6 = 0 p = 2 and p = 6

6. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: 4x2 36 = 0 (v) 4x2 36 = 0 (2x)2 (6)2 = 0 (2x 6)(2x + 6) = 0 2x 6 = 0 and 2x + 6 = 0 2x = 6 and 2x = 6 x = 3 and x = 3

7. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: a2 a 12 = 0 (vi) a2 a 12 = 0 (a 4)(a + 3) = 0 a 4 = 0 and a + 3 = 0 a = 4 and a = 3

8. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: (vii) 5k2 35k = 0 5k2 35k = 0 5k (k 7) = 0 5k = 0 and k 7 = 0 k = 0 and k = 7

9. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: (viii) b2 + 10b + 16 = 0 b2 + 10b + 16 = 0 (b + 2)(b + 8) = 0 b + 2 = 0 and b + 8 = 0 b = 2 and b = 8

10. 04 Practice Questions 4.3 1. Factorise and hence solve the following quadratic equations: x2 + 15x + 36 = 0 (ix) x2 + 15x + 36 = 0 (x + 3)(x + 12) = 0 x + 3 = 0 and x + 12 = 0 x = 3 x = 12

11. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 2c2 5c 12 = 0 (i) 2c2 5c 12 = 0 (2c + 3)(c 4) = 0 2c + 3 = 0 and c 4 = 0 2c = 3 and c = 4 c = -3 2

12. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 3x2 + 10x + 8 = 0 (ii) 3x2 + 10x + 8 = 0 (3x + 4)(x + 2) = 0 3x + 4 = 0 and x + 2 = 0 3x = 4 and x = 2 x = -4 3

13. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 11t2 32t 3 = 0 (iii) 11t2 32t 3 = 0 (11t + 1)(t 3) = 0 11t + 1 = 0 and t 3 = 0 11t = 1 and t = 3 t = -1 11

14. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 3a2 + 10a 8 = 0 (iv) 3a2 + 10a 8 = 0 (3a 2)(a + 4) = 0 3a 2 = 0 and a + 4 = 0 3a = 2 and a = 4 a =2 3

15. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 4x2 + 10x + 6 = 0 (v) 4x2 + 10x + 6 = 0 (2x + 3)(2x + 2) = 0 2x + 3 = 0 and 2x + 2 = 0 2x = 3 and 2x = 2 3 x = 2 and x = 1

16. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 4t2 + 12t + 9 = 0 (vi) 4t2 + 12t + 9 = 0 (2t + 3)(2t + 3) = 0 2t + 3 = 0 2t = 3 t = -3 one root only. 2

17. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: (vii) 4x2 23x + 15 = 0 4x2 23x + 15 = 0 (4x 3)(x 5) = 0 4x 3 = 0 and x 5 = 0 4x = 3 and x = 5 x =3 4

18. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: (viii) 9a2 + 12a + 4 = 0 9a2 + 12a + 4 = 0 (3a + 2)(3a + 2) = 0 3a + 2 = 0 3a = 2 a = - 2 one root only. 3

19. 04 Practice Questions 4.3 2. Factorise and hence solve the following quadratic equations: 12x2 + 32x + 5 = 0 (ix) 12x2 + 32x + 5 = 0 (6x + 1)(2x + 5) = 0 6x + 1 = 0 and 2x + 5 = 0 6x = 1 and 2x = 5 x = - 1 x = - 5 and 6 2

20. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. 4x2 46 = 3 (i) 4x2 46 = 3 subtract 3 from both sides 4x2 49 = 0 (2x)2 (7)2 = 0 (2x 7)(2x + 7) = 0 2x 7 = 0 and 2x + 7 = 0 2x = 7 and 2x = 7 x =7 x =-7 and 2 2

21. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. k2 7 = 9 (ii) k2 7 = 9 subtract 9 from both sides k2 16 = 0 (k)2 (4)2 = 0 (k 4)(k + 4) = 0 k 4 = 0 and k + 4 = 0 k = 4 and k = 4

22. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. 6x2 = 24x (iii) 6x2 = 24x subtract 24x from both sides 6x2 24x = 0 6x (x 4) = 0 6x = 0 and x 4 = 0 x = 0 and x = 4

23. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. 7x2 = 15x 2 (iv) 7x2 = 15x 2 subtract 15x from and add 2 to both sides 7x2 15x + 2 = 0 (7x 1)(x 2) = 0 7x 1 = 0 and x 2 = 0 7x = 1 and x = 2 x =1 7

24. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. 7k = 15 2k2 (v) 7k = 15 2k2 add 2k2 to and subtract 15 from both sides 2k2 + 7k 15 = 0 (2k 3)(k + 5) = 0 2k 3 = 0 and k + 5 = 0 2k = 3 and k = 5 k =3 2

25. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. 2x2 = 3x + 2 (vi) 2x2 = 3x + 2 subtract 3x and 2 from both sides 2x2 3x 2 = 0 (2x + 1)(x 2) = 0 2x + 1 = 0 and x 2 = 0 2x = 1 and x = 2 x = - 1 2

26. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. (vii) 9a2 = 73a 8 9a2 = 73a 8 subtract 73a from and add 8 to both sides 9a2 73a + 8 = 0 (9a 1)(a 8) = 0 9a 1 = 0 and a 8 = 0 9a = 1 and a = 8 a =1 9

27. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. (viii) 9h2 + 4 = 15h 9h2 + 4 = 15h subtract 15h from both sides 9h2 15h + 4 = 0 (3h 4)(3h 1) = 0 3h 4 = 0 and 3h 1 = 0 3h = 4 and 3h = 1 h =1 h =4 and 3 3

28. 04 Practice Questions 4.3 3. Rearrange the following quadratic equations into the form ax2 + bx + c = 0, and hence factorise and solve. 5x2 11x 3 = 2x + 3 (ix) 5x2 11x 3 = 2x + 3 subtract 2x and 3 from both sides 5x2 13x 6 = 0 (5x + 2)(x 3) = 0 5x + 2 = 0 and x 3 = 0 5x = 2 and x = 3 x = - 2 5

29. 04 Practice Questions 4.3 4. Use the quadratic formula to solve the following quadratic equations. Leave your answer in simplest surd form. x2 + 2x 2 = 0 (i) x2 + 2x 2 = 0 a = 1, b = 2, c = 2 x =-b b2-4ac 2a =-2 22-4(1)(-2) 2(1) =-2 2 3 2 =-2 4-(-8) =-1 3 2 =-2 12 x =-1+ 3 and x =-1- 3 2

30. 04 Practice Questions 4.3 4. Use the quadratic formula to solve the following quadratic equations. Leave your answer in simplest surd form. a2 + 6a 1 = 0 (ii) a2 + 6a 1 = 0 a = 1, b = 6, c = 1 a =-b b2-4ac 2a =-6 62-4(1)(-1) 2(1) =-6 2 10 2 =-6 36-(-4) =-3 10 2 =-6 40 a=-3+ 10 and a=-3- 10 2

31. 04 Practice Questions 4.3 4. Use the quadratic formula to solve the following quadratic equations. Leave your answer in simplest surd form. x2 7x 4 = 0 (iii) x2 7x 4 = 0 a = 1, b = 7, c = 4 x =-b b2-4ac 2a =-(-7) (-7)2-4(1)(-4) 2(1) =7 49-(-16) 2 x =7+ 65 and x =7- 65 =7 65 2 2 2

32. 04 Practice Questions 4.3 4. Use the quadratic formula to solve the following quadratic equations. Leave your answer in simplest surd form. p2 + 8p 4 = 0 (iv) p2 + 8p 4 = 0 a = 1, b = 8, c = 4 p=-b b2-4ac 2a =-8 4 5 =-8 82-4(1)(-4) 2(1) 2 p=-8 4 5 2 =-8 64-(-16) 2 =-4 2 5 =-8 80 p=-4+2 5 and p=-4-2 5 2

33. 04 Practice Questions 4.3 4. Use the quadratic formula to solve the following quadratic equations. Leave your answer in simplest surd form. 2x2 + 5x + 1 = 0 (v) 2x2 + 5x + 1 = 0 a = 2, b = 5, c = 1 x =-b b2-4ac 2a =-5 52-4(2)(1) 2(2) =-5 25-8 4 x =-5+ 17 and x =-5- 17 =-5 17 4 4 4

34. 04 Practice Questions 4.3 4. Use the quadratic formula to solve the following quadratic equations. Leave your answer in simplest surd form. 3k2 10k + 5 = 0 (vi) 3k2 10k + 5 = 0 a = 3, b = 10, c = 5 k =-b b2-4ac 2a =10 2 10 =-(-10) (-10)2-4(3)(5) 2(3) 6 =10 100-60 =5 10 6 3 k =5+ 10 and k =5- 10 =10 40 3 3 6

35. 04 Practice Questions 4.3 5. Rearrange the following quadratic equations into the form ax2 + bx + c = 0 and hence solve using the quadratic formula. Give your answer to two decimal places. x2 + 5x = 1 (i) x2 + 5x = 1 x2 + 5x + 1 = 0 a = 1, b = 5, c = 1 =-5 21 x =-b b2-4ac 2 2a x =-5+ 21 and x =-5- 21 =-5 52-4(1)(1) 2(1) 2 2 =-5 25-4 x=-0 21 and x=-4 79 2

36. 04 Practice Questions 4.3 5. Rearrange the following quadratic equations into the form ax2 + bx + c = 0 and hence solve using the quadratic formula. Give your answer to two decimal places. x2 = 3x + 1 (ii) x2 = 3x + 1 x2 3x 1 = 0 a = 1, b = 3, c = 1 =3 13 x =-b b2-4ac 2 2a x =3+ 13 and x =3- 13 =-(-3) (-3)2-4(1)(-1) 2(1) 2 2 =3 9-(-4) x=3 30 and x=-0 30 2

37. 04 Practice Questions 4.3 5. Rearrange the following quadratic equations into the form ax2 + bx + c = 0 and hence solve using the quadratic formula. Give your answer to two decimal places. 9a = 4 a2 (iii) 9a = 4 a2 a2 + 9a 4 = 0 a = 1, b = 9, c = 4 =-9 97 a =-b b2-4ac 2 2a x =-9+ 97 and x =-9- 97 =-9 92-4(1)(-4) 2(1) 2 2 =-9 81-(-16) x=0 42 and x=-9 42 2

38. 04 Practice Questions 4.3 5. Rearrange the following quadratic equations into the form ax2 + bx + c = 0 and hence solve using the quadratic formula. Give your answer to two decimal places. k2 4 = 7k (iv) k2 4 = 7k k2 7k 4 = 0 a = 1, b = 7, c = 4 =7 65 k =-b b2-4ac 2 2a k =7+ 65 and k =7- 65 =-(-7) (-7)2-4(1)(-4) 2(1) 2 2 =7 49-(-16) k =7 53 and k =-0 53 2

39. 04 Practice Questions 4.3 5. Rearrange the following quadratic equations into the form ax2 + bx + c = 0 and hence solve using the quadratic formula. Give your answer to two decimal places. 8x = 2 5x2 (v) 8x = 2 5x2 5x2 + 8x 2 = 0 a = 5, b = 8, c = 2 =-8 104 10 x =-b b2-4ac 2a x =-8+ 104 and x =-8- 104 =-8 82-4(5)(-2) 2(5) 10 10 =-8 64-(-40) 10 x=0 22 and x=-1 82

40. 04 Practice Questions 4.3 5. Rearrange the following quadratic equations into the form ax2 + bx + c = 0 and hence solve using the quadratic formula. Give your answer to two decimal places. 4p2 = 3 5p (vi) 4p2 = 3 5p 4p2 + 5p 3 = 0 a = 4, b = 5, c = 3 =-5 73 x =-b b2-4ac 8 2a x =-5+ 73 and x =-5- 73 =-5 52-4(4)(-3) 2(4) 8 8 =-5 25-(-48) x=0 44 and x=-1 69 8

41. 04 Practice Questions 4.3 6. The function f (x) = x2 x 6. Solve the equation f (x) = 0. (i) f(x) = x2 x 6 = 0 (x 3)(x + 2) = 0 x 3 = 0 and x + 2 = 0 x = 3 and x = 2

42. 04 Practice Questions 4.3 6. (ii) Hence draw a sketch of the graph of f (x). Roots are 2 and 3, so the graph crosses the x-axis at these values f (x) = x2 x 6 has a positive x2 term, so the graph is U shaped Let x = 0: f(0) = 02 0 6 y = 6 (0, 6) is on the graph Sketch:

43. 04 Practice Questions 4.3 7. The function g(x) = x2 + 3x + 4. Solve the equation g(x) = 0. (i) g(x) = x2 + 3x + 4 = 0 (multiply all parts by 1) x2 3x 4 = 0 (x 4)(x + 1) = 0 x 4 = 0 and x + 1 = 0 x = 4 and x = 1

44. 04 Practice Questions 4.3 7. (ii) Hence draw a sketch of the graph of g(x). Roots are 1 and 4, so the graph crosses the x-axis at these values g(x) = x2 + 3x + 4 has a negative x2term, so the graph is shaped Let x = 0 g(0) = (0)2 + 3(0) + 4 g = 4 (0, 4) is on the graph. Sketch:

45. 04 Practice Questions 4.3 8. The function (x) = 2x2 3x 5. Solve the equation (x) = 0. (i) h(x) = 2x2 3x 5 = 0 (2x 5)(x + 1) = 0 2x 5 = 0 and x + 1 = 0 2x = 5 and x = 1 x = 2 5

46. 04 Practice Questions 4.3 8. (ii) By replacing (x) with 2x2 3x 5, express (x) = 9 in terms of x. h(x) = 2x2 3x 5 = 9 2x2 3x 14 = 0

47. 04 Practice Questions 4.3 8. (iii) Hence, solve the equation h(x) = 9. 2x2 3x 14 = 0 (2x 7)(x + 2) = 0 2x 7 = 0 and x + 2 = 0 2x = 7 and x = 2 x = 3 5

48. 04 Practice Questions 4.3 8. (iv) Draw a sketch of the graph of (x). Roots are 1 and 2 5, so the graph crosses the x-axis at these values (x) = 2x2 3x 5 has a positive x2 term, so the graph is U shaped Let x = 0: h(0) = 2(0)2 3(0) 5 = 5 y = 5 (0, 5) is on the graph.

49. 04 Practice Questions 4.3 9. The function f (x) = 3x2 + 2x + 8. Solve the equation f (x) = 0. (i) f(x) = 3x2 + 2x + 8 = 0 (multiply all parts by 1) 3x2 2x 8 = 0 (3x + 4)(x 2) = 0 3x + 4 = 0 and x 2 = 0 3x = 4 and x = 2 x = - 4 3

50. 04 Practice Questions 4.3 9. (ii) By replacing f(x) with 3x2 + 2x + 8, express f (x) = 3 in terms of x. 3x2 + 2x + 8 = 3 3x2 + 2x + 5 = 0

Load More ...