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20
New Trend Mathematics S4A — Supplement
Chapter 2
Quadratic Equations
WARM - UP E XERCISE
Expand the following expressions. (1  3)
1. (a) (x  5) 2
(b) (2x  3) 2
2. (a) (x  4) 2
(b) (3x  7) 2
3. (a) (x  9)(x  9)
(b) (4x  11)(4x  11)
Factorize the following expressions. (4  11)
4. (a) x 2  6x
(b) 4x 2  12x
5. (a) (x  1) 2  (x  1)
(b) (2x  5) 2  9(2x  5)
6. (a) x 2  4x  4
(b) x 2  12x  36
7. (a) x 2  14x  49
(b) x 2  26x  169
8. (a) x 2  4
(b) 9x 2  16
9. (a) x 2  8x  7
(b) 4x 2  23x  15
10. (a) x 2  10x  16
(b) 6x 2  35x  36
11. (a) x 2  3x  108
(b) 15x 2  2x  8
Solve the following inequalities. (12  16)
12. (a) x  4 > 6
(b) 7  x  1
13. (a) 3  5x  28
(b) 9 < 4x  5
14. (a) 7x > 2x  5
(b) 4x > 2(3x  5)  1
15. (a) 3x  (2x  5)  x
(b) 4x  (7x  2)  3x
16. (a) 3(2  4x)  14  3x
(b) 4  3x  6(x  4)  x
Chapter 2 Quadratic Equations
21
B UILD - UP E XERCISE
[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set,
Intermediate Set and Advanced Set. You may choose to c omplete any ONE set according to your need. ]
Exercise 2A
  El em en tar y S et
 
Level 1
Rewrite each of the following quadratic equations into ax 2  bx  c  0 where a  0. Write
down the values of a, b and c. (1  4)
1. x 2  9  2x
2. 9x 2  5x  4
3. 3x 2  7x
4. 3  2x 2  6x
Solve the following equations. (5  10)
5. x(x  1)  0
7. (x  2)(x  4)  0
9. (2x  7)(5x  1)  0
6. x(3x  1)  0
8. (x  3)(x  10)  0
10. (3x  2)(4x  7)  0
11. x 2  7x  0
12. x 2  9  0
13. x 2  8x  15  0
14. x 2  x  12  0
15. x 2  16x  64  0
16. x 2  17x  60  0
17. 2x 2  7x  9  0
18. 2x 2  9x  4  0
19. 25x 2  20x  4  0
20. 75x 2  12  0
21. 18x 2  2x  0
22. 10  40x 2  0
23. 48  14x  x 2  0
24. 2x 2  9x  35
25. 33x  5x 2  18
Level 2
Solve the following equations by factorization. (26  34)
14  x
26. x 2 
27. 4(x 2  1)  17x
3
13
28. 4x(8x  3)  27
29. x 2  1  x
6
30. (x  8)(x  12)  4
31. x(2x  7)  3x(x  1)  0
32. (x  4)(x  9)  2(x  6)  0
34. 3x(3x  1)  2(3x  1)
33. (x  3) 2  (x  3)  0
Ex.2A Elementary Set
Solve the following equations by factorization. (11  25)
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New Trend Mathematics S4A — Supplement
 Intermediate Set
Level 1

Rewrite each of the following quadratic equations into ax 2  bx  c  0 where a  0. Write
down the values of a, b and c. (35  38)
35. x 2  16  3x
36. 7x 2  4x  10
37. 4x 2  9x
38. 2  5x 2  8x
Solve the following equations. (39  41)
39. x(x  3)  0
40. (x  4)(x  7)  0
41. (2x  1)(x  3)  0
Solve the following equations by factorization. (42  55)
Ex.2A Intermediate Set
42. 4x 2  49  0
43. x 2  2x  15  0
44. x 2  18x  72  0
45. x 2  22x  121  0
46. 3x 2  10x  8  0
47. 6x 2  5x  4  0
48. 12x 2  35x  8  0
49. 49x 2  42x  9  0
50. 98x 2  288  0
51. 27x 2  126x  0
52. 144  100x 2  0
53. 6  13x  2x 2  0
54. 8x 2  22x  5
55. 41x  18x 2  10
Level 2
Solve the following equations by factorization. (56  66)
33  4 x
56.
57. 5(x 2  2)  27x
 x2
5
29
58. 27  2x(15  4x)
59. x 2  1   x
10
60. (x  13)(x  19)  9
61. 4x(2  x)  5x(2x  3)  0
62. (3x  7)(3x  2)  6(3x  4)  0
63. (x  6) 2  4(x  6)  0
64. 4x(2x  1)  7(1  2x)
65. (5x  11) 2  (5x  11)(2x  1)
66. x(10x  3)  (5x  3)(3x  4)
Chapter 2 Quadratic Equations
23
 Advanced Set

Level 1
Solve the following equations. (67  69)
67. x(2x  1)  0
68. (3  x)(x  2)  0
69. (4  x)(5  2x)  0
70. 3x 2  192  0
71. 3x 2  14x  8  0
72. 5x 2  6x  8  0
73. 20x 2  84x  27  0
74. 36x 2  132x  121  0
75. 324x 2  100
76. 55x 2  20x
77. 567  175x 2  0
78. 21  5x  6x 2  0
79. 60x 2  10  x
80. 11x  15  12x 2
Level 2
Solve the following equations by factorization. (81  94)
 47 x  30
81.
82. 7(x 2  3)  46x
 x2
7
1 8
83. 35  4x(3x  13)
84. x 2   x
4 15
85. (3x  5)(3x  17)  36
86. 7x(2  3x)  2x(3  5x)  0
87. (4x  5)(4x  3)  2(7x  11)  0
88. (3x  5) 2  9(3x  5)  0
89. 9x(1  4x)  2(4x  1)
90. (4x  9) 2  (4x  9)(6x  13)
91. (2x  1)(x  8)  x(25  x)
92. (9x  10)(2x  3)  x(11x  6)
93. (2x  5) 2  (x  4) 2
94. (3x  2) 2  (2x  3) 2
95. Solve the following equations where p and q are real constants. Express the answers in
terms of p and q.
(a) x 2  p 2  q 2  2pq  0
(b) x 2  (p  q)x  (p  1)(q  1)  0
Ex.2A Advanced Set
Solve the following equations by factorization. (70  80)
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New Trend Mathematics S4A — Supplement
Exercise 2B
  El em en tar y S et
 
Level 1
Find the value of a of each of the following identities. (1  4)
1. x 2  14x  a  (x  7) 2
2. x 2  18x  a  (x  9) 2
3. 2x 2  4x  2  2(x  a) 2
4. 2x 2  ax  8  2(x  2) 2
Solve the following equations. (5  20)
5. (x  5) 2  4
6. (x  1) 2  4
7. (3x  5) 2  4
8. (2x  5) 2  9
1
9. ( x  ) 2  4
2
1
10. ( x  ) 2  25
3
Ex.2B Elementary Set
11. 36(x  2) 2  1
12. 4(x  6) 2  49
13. 9(x  5) 2  25
14. 64(3x  1) 2  4
15. (x  6) 2  13
16. (3x  5) 2  32
17. (2x  7) 2  18
18. (17  6x) 2  245
19. 4(x  5) 2  44
x
20. 4(  5) 2  21
2
Level 2
Solve the following equations by completing the square. (21  31)
21. x 2  4x  12  0
22. x 2  3x  40  0
23. x 2  10x  27  0
24. x 2  6x  15  0
25. x 2  7x  11  0
26. 2x 2  x  7  0
27. 3x 2  x  1  0
28. 2x 2  3  5x
29. 4(3x  1)  9x 2
30.
31.
1 2
x  4x  7  0
3
1
x  5  x2
2
Chapter 2 Quadratic Equations
25
 Intermediate Set

Level 1
Find the value of a of each of the following identities. (32  35)
32. x 2  20x  a  (x  10) 2
33. x 2  30x  a  (x  15) 2
34. 4x 2  16x  16  4(x  a) 2
35. 5x 2  ax  45  5(x  3) 2
36. (x  2) 2  16
37. (x  5) 2  9
38. (5x  4) 2  16
39. (4x  3) 2  25
3
40. ( x  ) 2  9
5
3
41. ( x  ) 2  36
4
42. 50(x  3) 2  2
43. 9(2x  3) 2  16
44. 16(3x  4) 2  49
45. (x  3) 2  20
46. (5x  7) 2  80
47. (4x  11) 2  75
48. 3(5x  1) 2  36
x
49. 9(  1) 2  26
4
Level 2
Solve the following equations by completing the square. (50  62)
50. x 2  6x  16  0
51. x 2  5x  84  0
52. x 2  12x  40  0
53. x 2  10x  36  0
54. x 2  13x  21  0
55. 3x 2  7x  3  0
56. 5x 2  2x  3  0
57. 3x 2  10x  8
58. 4x 2  5(4x  5)
59. x 2  2 
60.
1 2
x  5x  11  0
4
62. 2(x  3) 2  6x  17
3
x
2
61. x 2  4 5 x  6  0
Ex.2B Intermediate Set
Solve the following equations. (36  49)
26
New Trend Mathematics S4A — Supplement
 Advanced Set
Level 1
Solve the following equations. (63  76)
x
63. (x  4) 2  16
64. (  2) 2  9
3

65. (3x  7) 2  64
66. (7x  5) 2  36
5
67. ( x  ) 2  16
6
5
68. ( x  ) 2  49
8
69. 50(4x  1) 2  72
70. 108(5x  2) 2  147
71. (x + 10) 2  45
72. (4x  9) 2  150
73. 4(3x  7) 2  80
x
74. 16(  3) 2  125
7
75. ( x  5 ) 2  11
76. ( 3 x  4) 2  9
Level 2
Ex.2B Advanced Set
Solve the following equations by completing the square. (77  91)
77. x 2  8x  33  0
78. x 2  9x  52  0
79. x 2  16x  72  0
80. x 2  14x  42  0
81. x 2  11x  14  0
82. 4x 2  5x  2  0
83. 6x 2  7x  3  0
84. 5x 2  19x  12
85. 3(8x  3)  16x 2
86.
87.
1 2
x  6x  9  0
5
89. x 2  2 3 x  4  0
8
x  1  x2
3
88. x 2  6 7 x  20  0
90. 3(x  1) 2  20x  9
91. 5(x  2) 2  2(8  5x)
92. Consider the equation x 2  4px  2q  0 where p and q are real constants.
(a) By completing the square, show that (x  2p) 2  4p 2  2q.
(b) Hence show that x  2 p  4 p 2  2q .
(c) Use the above results to solve the following equations.
(i) x 2  8x  14  0
(ii) x 2  16x  6  0
Chapter 2 Quadratic Equations
27
Exercise 2C
1. x 2  3x  2  0
2. x 2  6x  7  0
3. x 2  6x  9  0
4. x 2  8x  12  0
5. 2x 2  5x  3  0
6. 3x 2  26x  16  0
7. 9x 2  42x  49  0
8. 25x 2  30x  9  0
9. x 2  2x  9  0
10. 5x 2  4x  2  0
11. 2x 2  4x  5  0
12. 4x 2  5  3x
13. 9x  2(x 2  3)
14. x 2  4(x  2)
15. 3(x 2  1)  14(2  x)
16. 4x(x  3)  20  x
Ex.2C Elementary Set
  El em en tar y S et
 
Level 1
Solve the following equations by using the quadratic formula. (1  16)
Level 2
Solve the following equations by using the quadratic formula. (17  20)
x 2  11
x2 x 3
 
17. x 
18.
8
2 5 5
19. x 2  2 5 x  3  0
20. (x  3)(x  3)  6x  1
21. x 2  3x  10  0
22. 3x 2  5x  2  0
23. 5x 2  27x  10  0
24. 16x 2  56x  49  0
25. 121x 2  132x  36  0
26. x 2  4x  9  0
27. 7x 2  12x  3  0
28. 3x 2  9x  8  0
29. 6x 2  7  8x
30. 10x  3(x 2  1)
31. 2x 2  3(4x  1)
32. 6x(x  4)  42  5x
Ex.2C Intermediate Set
 Intermediate Set

Level 1
Solve the following equations by using the quadratic formula. (21  32)
28
New Trend Mathematics S4A — Supplement
Ex.2C Intermediate Set
Level 2
Solve the following equations by using the quadratic formula. (33  40)
1 x2 x
x2  3

33. x 
34.
10
3
2
35.
x2 x 1
 
4 6 3
36. 2 
2 x 2 3x

5
4
37. 2 x 2  3 7 x  1  0
38. (2x  1)(2x  1)  1  x
39. (4x  5) 2  6x(2x  3)
40. (3x  4)(x  6)  5(2x  1)  1
 Advanced Set

Level 1
Solve the following equations by using the quadratic formula. (41  50)
41. 4x 2  19x  5  0
42. 6x 2  31x  30  0
43. 162x 2  72x  8  0
44. 64x 2  144x  81  0
45. x 2  6x  6  0
46. 10x 2  22x  9  0
47. 5x 2  8x  4  0
48. 3  5x  9x 2
49. 6x  5(x 2  3)
50. 11x  35  6x(x  6)
Ex.2C Advanced Set
Level 2
Solve the following equations by using the quadratic formula. (51  58)
x 2  23
x 2  12
 x
51. x 
52.
6
14
53.
x2 2x
1


4
5
15
54. 4 x  3x 2 
4
3
55. 5x 2  4 3x  3  0
56. (4 x  1)(4 x  1)  10 3x  4
57. (5x  1) 2  8x(2x  3)
58. (4 x  5)( x  2)  3(3x  7)  1
59. (a) Solve 30x 2  x  14  0 by using the quadratic formula.
(b) Hence solve 30(x  2) 2  (x  2)  14  0.
60. A root of the quadratic equation ax 2  20x  3  0 is 3.
(a) Find the value of a.
(b) Hence find the other root of the equation by using the quadratic formula.
Chapter 2 Quadratic Equations
29
Exercise 2D
  El em en ta r y S et
 
Level 1
1. Find the value of the discriminant of each of the following quadratic equations.
(a) x 2  6x  4  0
(b) x 2  3x  10  0
(c) 2x 2  6x  5  0
(d) 3x 2  5x  4  0
(e) x 2  8x  3
(f) 4x 2  25  20x
2. Determine the nature of the roots of each of the following quadratic equations.
(a) x 2  x  4  0
(b) 2x 2  6x  9  0
(c) 4x 2  36x  81  0
(d) 2x 2  9x  5
3. (a) Find the discriminant of the quadratic equation x 2  8x  k  0.
(b) Find the range of values of k if the equation x 2  8x  k  0 has real roots.
4. (a) Find the discriminant of the quadratic equation 3x 2  16x  4k  0.
5. (a) Find the discriminant of the quadratic equation kx 2  12x  4  0.
(b) Find the range of values of k if the equation kx 2  12x  4  0 has no real roots.
6. (a) Find the discriminant of the quadratic equation 5kx 2  24x  6  0.
(b) Find the range of values of k if the equation 5kx 2  24x  6  0 has no real roots.
7. (a) Find the discriminant of the quadratic equation 2x 2  8x  k  1  0.
(b) Find the value of k if the equation 2x 2  8x  k  1  0 has two equal real roots.
8
8. (a) Find the discriminant of the quadratic equation 4 x 2  x  2  k  0 .
3
8
(b) Find the value of k if the equation 4 x 2  x  2  k  0 has two equal real roots.
3
9. Find the value of k if each of the following quadratic equations has two equal real roots.
(a) kx 2  10x  1  0
(b) 3x 2  8x  4k  0
10. Find the range of values of k if each of the following quadratic equations has two unequal
real roots.
(a) 2x 2  x  k  0
(b) kx 2  2x  4  0
11. Find the range of values of k if each of the following quadratic equations has no rea l roots.
(a) x 2  9x  k  0
(b) kx 2  4x  5  0
Ex.2D Elementary Set
(b) Find the range of values of k if the equation 3x 2  16x  4k  0 has real roots.
30
New Trend Mathematics S4A — Supplement
Level 2
Ex.2D Elementary Set
12. Find the range of values of k if the quadratic equation x 2  4x  k  1  0 has two unequal
real roots.
13. Find the range of values of k if the quadratic equation 2x(x  3)  x  k has no real roots.
14. Find the range of values of k if the quadratic equation x 2  5x  3k  1  0 has real roots.
15. Find the values of k if the quadratic equation x 2  kx  4k  0 has a double root.
 Intermediate Set

Level 1
16. Find the value of the discriminant of each of the following quadratic equations.
(a) x 2  8x  2  0
(b) x 2  6x  3  0
(c) 3x 2  7x  2  0
(d) 4x 2  3x  11  0
(e) 2x 2  5x  6
(f) 24x  16x 2  9
17. Determine the nature of the roots of each of the following quadratic equations.
(a) x 2  10x  25  0
(b) 2x 2  x  5  0
(c) 4x 2  x  8  0
(d) 3  2x  5x 2
18. (a) Find the discriminant of the quadratic equation 2x 2  10x  k  0.
Ex.2D Intermediate Set
(b) Find the range of values of k if the equation 2x 2  10x  k  0 has real roots.
19. (a) Find the discriminant of the quadratic equation 3kx 2  18x  8  0.
(b) Find the range of values of k if the equation 3kx 2  18x  8  0 has no real roots.
20. (a) Find the discriminant of the quadratic equation 2x 2  6x  (k  2)  0.
(b) Find the value of k if the equation 2x 2  6x  (k  2)  0 has two equal real roots.
21. Find the value of k if each of the following quadratic equations has two equal real roots.
(a) 4kx 2  60x  25  0
(b) 5x 2  16x  8k
22. Find the range of values of k if each of the following quadratic equations has two unequal
real roots.
(a) x 2  3x  4k  0
(b) kx 2  5x  2  0
23. Find the range of values of k if each of the following quadratic equations has no real roots.
(a) x 2  14x  6k  0
(b) kx 2  6x  15  0
Level 2
24. Find the range of values of k if the quadratic equation 5x 2  3x  2(k  1)  0 has two
unequal real roots.
Chapter 2 Quadratic Equations
31
26. Find the range of values of k if the quadratic equation 3(2x 2  1)  2(4x  k) has real roots.
27. Find the values of k if the quadratic equation x 2  2kx  k  2  0 has a double root.
28. (a) Find the value of k if the quadratic equation 3kx 2  kx  2  0 has a double root.
(b) Using the result of (a), solve the equation 3kx 2  kx  2  0.
29. (a) Find the value of k if the quadratic equation kx 2  (3k  1)x  (2k  1)  0 has a double
root.
Ex.2D Intermediate Set
25. Find the range of values of k if the quadratic equation 5x  4  3(x 2  k) has no real roots.
(b) Using the result of (a), solve the equation kx 2  (3k  1)x  (2k  1)  0.
 Advanced Set

Level 1
30. Find the value of the discriminant of each of the following quadratic equations.
(a) x 2  2x  12  0
(b) 2x 2  7x  4  0
(c) 5x 2  6x  1  0
(d) 6x 2  2x  5  0
(e) 3x  2x 2  2
(f) 42x  9  49x 2
(a) x 2  2x  4  0
(b) 3x 2  6x  3  0
(c) 7x 2  4x  5  0
(d) 5  11x  6x 2
32. Find the value of k if each of the following quadratic equations has two equal real roots.
(a) kx 2  7x  28  0
(b) 10x 2  12x  3k
33. Find the range of values of k if each of the following quadratic equations has two unequal
real roots.
(a) 3x 2  4x  2k  0
(b) 2kx 2  3x  4  0
34. Find the range of values of k if each of the following quadratic equations has no real roots.
(a) x 2  12k  18x
(b) kx 2  10  16x
Level 2
35. Find the range of values of k if the quadratic equation (k  1)x 2  5x  2  0 has two
unequal real roots.
36. Find the range of values of k if the quadratic equation (k  1)x 2  12  4x has no real
roots.
37. Find the range of values of k if the quadratic equation k(1  2x)  3  2x  kx 2 has real
roots.
Ex.2D Advanced Set
31. Determine the nature of the roots of each of the following quadratic equations.
32
New Trend Mathematics S4A — Supplement
38. Find the values of k if the quadratic equation k(x 2  1)  6x has a double root.
39. (a) Find the value of k if the quadratic equation 5kx2  3kx 
Ex.2D Advanced Set
(b) Using the result of (a), solve the equation 5kx2  3kx 
1
 0 has a double root.
2
1
 0.
2
40. (a) Find the value of k if the quadratic equation 2kx 2  (4k  3)x  (2k  5)  0 has a
double root.
(b) Using the result of (a), solve the equation 2kx 2  (4k  3)x  (2k  5)  0.
41. (a) Express the discriminant of the quadratic equation 4 kx 2  (k  8)x  2  0 in terms of k.
(b) Hence prove that the quadratic equation 4kx 2  (k  8)x  2  0 has real roots for all
real values of k.
Exercise 2E
  El em en tar y S et
Level 1
Solve the following equations. (1  10)
 
Ex.2E Elementary Set
1. (x  1) 2  2(x  1)  3  0
2. (x  4) 2  7(x  4)  18  0
x
x
3. (  1) 2  2(  1)  8  0
2
2
4. 6(x  2) 2  5(x  2)  1  0
3
3
5. ( ) 2  7( )  6  0
x
x
4
4
6. 2( ) 2  5( )  3  0
x
x
7. (
1 2
1
)  5(
)  24  0
x 1
x 1
9. ( x ) 2  4 x  3  0
8. (x 2 ) 2  3(x 2 )  54  0
10. (3 x ) 2  6(3 x )  27  0
Level 2
Solve the following equations. (11  15)
11. 3x  2 x  1  0
13.
2
5
x 1


x  4 3( x  4)
12
15.
1
1
6


x3 x3 5
12. x 4  26x 2  25  0
14.
2
4 1
 
x3 x 5
Chapter 2 Quadratic Equations
 Intermediate Set
Level 1
Solve the following equations. (16  23)
18. (
1 2
1
)  5(
)  14  0
x2
x2

17. 2(3x  1) 2  3(3x  1)  5  0
19. 4(
1 2
1
)  4(
) 30
3 x
3 x
20. (x 2 ) 2  2(x 2 )  48  0
21. 3( x ) 2  5 x  2  0
22. (2 x ) 2  5(2 x )  4  0
23. (4 x ) 2  20(4 x )  64  0
Level 2
Solve the following equations. (24  32)
24. 2x  13 x  15  0
25. x 4  22x 2  72  0
26.
1
3
2x  1


x  2 2( x  2)
8
27.
7
2
x4


2(3  x) 3  x
4
28.
3 5
6
 
4 x x 1
29.
1
1
4


x  2 x  2 15
2
30. ( x  6)(5  )  9
x
32. (1 
Ex.2E Intermediate Set
16. (2x  3) 2  2(2x  3)  8  0
33
4
31. (3  )( x  3)  5
x
20 8
)(  1)  3
x x
35. 3(
3 2
3
)  2(
)50
2x  1
2x  1
37. 2( x ) 2  5 x  12  0
36. 4(x 2 ) 2  11(x 2 )  7  0
38. 2(2 2x )  5(2 x )  2  0
Level 2
Solve the following equations. (39  47)
39. 12 x  11 x  2  0
41.
11
5
3 x


4(2 x  1) 2 x  1
8
40. 2x 4  23x 2  45  0
42.
x
3
1

x
5( x  3) 2( x  3)
2
Ex.2E Advanced Set
 Advanced Set

Level 1
Solve the following equations. (33  38)
x
x
33. (1  5x) 2  5(1  5x)  4  0
34. 4(  1) 2  7(  1)  2  0
4
4
34
43.
New Trend Mathematics S4A — Supplement
3
2
4
 
x3 x
3
44.
4
45. (6 x  1)(  1)  18
x
47.
1
1
12


3x  2 3x  2 7
2
4
46. 7  (  4)(1  )
x
x
x 18
  x 1
3 x
48. (a) Solve the equation y 2  2y  3  0.
Ex.2E Advanced Set
(b) Hence solve the equation (x 2  2x) 2  2(x 2  2x)  3  0.
49. (a) Solve the equation 4t 2  21t  26  0.
3
3
(b) Hence solve the equation 4( x  ) 2  21( x  )  26  0 .
x
x
2
 7 y  15 .
y
2
 7 x  5  15 .
(b) Hence solve the equation
x5
50. (a) Solve the equation
3 2x  3
…………. (1).
4
(a) Let u  2 x  3 . Show that 2u 2  3u  2  0………… (2).
(b) Solve equation (2) and hence solve equation (1).
51. Consider the equation x  2 
Exercise 2F
  El em en tar y S et
 
Level 1
1. The height of a moving balloon above the ground after x seconds is given by
(14  24 x  5 x 2 ) m . When is the balloon 50 m above the ground?
Ex.2F Elementary Set
2. The price of a plate is $(34  11x  x 2 ) where x cm is the radius of the plate. If the price of
the plate is $46, what is its radius?
3. The cost of making a bookshelf is $(120  0.45 x  0.15 x 2 ) where x cm is the height of the
bookshelf. Find the height of the bookshelf if the cost is $687.
4. In the figure, the area of rectangle ABCD is 36 cm 2 . Find the value of x.
A
(x  5) cm
D
x cm
B
C
Chapter 2 Quadratic Equations
35
5. The product of two consecutive numbers is 156.
(a) If the smaller number is x, find the larger number in terms of x.
(b) Find the two numbers.
6. The product of two consecutive odd numbers is 143.
(a) If the smaller number is x, find the larger number in terms of x.
(b) Find the two numbers.
7. The sum of two numbers is 30.
(a) If one of the numbers is x, find the other number in terms of x.
(b) Given that the product of the two numbers is 216, find the two numbers.
8. The difference between two numbers is 4.
(a) If the larger number is x, find the smaller number in terms of x.
(b) Given that the product of the two numbers is 480, find the two numbers.
9. In the figure, the sum of the areas of square ABCD and square PQRS is 130 cm 2 . Find the
value of x.
x cm
B
P (x  2) cm S
D
C
Q
R
n(n  1)
where n is a positive integer.
2
If 1  2  3    n  300, find n.
10. It is given that 1  2  3    n 
11. The perimeter of a rectangle is 30 cm.
(a) If the length of the rectangle is x cm, find its width in terms of x.
(b) Given that its area is 54 cm 2 , find the dimensions of the rectangle.
12. In the figure, the rectangular photo frame is 40 cm long and 24 cm wide. The border of the
frame is made of wood with width x cm. If the area of the photo displayed is 512 cm 2 , find
the value of x.
40 cm
x cm
x cm
x cm
24 cm
Photo
x cm
13. In the figure, ABC is a right-angled triangle. AB  x cm, BC  (2x  1) cm and
AC  (3x  7) cm. Find the value of x.
A
(3x  7) cm
C
(2x  1) cm
x cm
B
Ex.2F Elementary Set
A
36
New Trend Mathematics S4A — Supplement
14. It is given that a rectangular paper is 10 cm long and 8 cm wide. Four squares with sides
x cm are cut away from the four corners of the paper (as shown in the figure) in order to
make an open rectangular box with base area 35 cm 2 . Find the value of x.
x cm
x cm
8 cm
Ex.2F Elementary Set
10 cm
Level 2
15. John is 2 years older than his brother Tom, but 28 years younger than his mother Jenny.
Three years later, Jenny’s age will be exactly the product of her sons’. How old is Jenny
now?
16. A man bought some toy cars for $2 200 but twelve of them were broken. He sold each of
the remaining toy cars at $8 more than the cost. Finally, he obtained a profit of $440. Let
$x be the cost of each toy car.
(a) Express the number of toy cars sold in terms of x.
(b) Find the cost of each toy car.
17. A fast food restaurant charges $480 to a group of students for the food of a Christmas
party. If 10 students cannot join the party, each of the remaining students has to pay $4
more to cover the charge. Find the original number of students who planne d to join the
party.
 Intermediate Set
Level 1

18. The distance between a coastline and a ship after x hours is given by (12  7 x  3x 2 ) km .
When are the ship and the coastline 52 km apart?
19. A ball is thrown vertically upwards. Its height above the ground after x seconds is given by
(1  7 x  2 x 2 ) m . When is the ball 4 m above the ground?
Ex.2F Intermediate Set
20. The sum of two numbers is 42.
(a) If one of the numbers is x, find the other number in terms of x.
(b) Given that the product of the two numbers is 392, find the two numbers.
21. The difference between two numbers is 7.
(a) If the smaller number is x, find the larger number in terms of x.
(b) Given that the product of the two numbers is 450, find the two n umbers.
n(n  1)
where n is a positive integer.
2
(a) If 1  2  3    n  630, find n.
(b) If 31  32  33    n  1 488, find n.
22. It is given that 1  2  3    n 
Chapter 2 Quadratic Equations
37
23. A square with sides 4 cm is cut away from each corner of a tin sheet with di mensions
x cm  (x  15) cm. Then the sheet is folded up to form an open box. If the capacity of the
box is 504 cm3 , find the length and the width of the box.
4 cm
4 cm
x cm
(x + 15) cm
24. After cutting a small rectangle with width x cm from a corner of a large rectangle with an
area of 84 cm 2 , two of the sides of the remaining figure become 10 cm and 4 cm as shown.
If the length of the small rectangle is 2 times its width, find the dimensions of the small
rectangle.
10 cm
x cm
25. In the figure, ABCD is a trapezium where CD  AD. If the area of ABCD is 176 cm 2 , find
the length of AD.
C (2x  3) cm B
(x  2) cm
D
(4x  5) cm
A
26. The product of two consecutive even numbers is 224. Find the two numbers.
Level 2
27. The sum of the reciprocals of two consecutive even numb ers is
23
. Find the two numbers.
264
28. Find the dimensions of a rectangle with a perimeter of 45 m and an area of 116 m 2 .
29. Owen is 28 years younger than his mother and 35 years younger than his father. Five years
later, the sum of the ages of Owen’s parents will be exactly the square of the age of Owen.
How old is Owen now?
30. 84 students are evenly divided into groups. If the number of groups decreases by 3, the
number of students in each group increases by 9. What is the original number of groups?
31. A man bought some calculators for $4 600 but four of them were broken. He sold each of
the remaining calculators at $40 more than the cost. Finally, he obtained a profit of $980.
Find the cost of each calculator.
Ex.2F Intermediate Set
4 cm
38
New Trend Mathematics S4A — Supplement
Ex.2F Intermediate Set
32. Sammy had bought a number of compact discs from a shop for $40 last week. The shop
reduced the price of each compact disc by $0.4 this week. If Sammy bought the compact
discs this week, she would buy 10 more compact discs by paying extra $2. What was the
original price of each compact disc?
33. A company charges $600 to a group of students for the rental of a coach for a picnic. If
5 students cannot join the picnic, each of the remaining students has to pay $4 more to
cover the charge. Find the original number of students who planned to join the picnic.
 Advanced Set

Level 1
34. The cost of making a model is $(140  0.55 x  0.2 x 2 ) where x cm is the height of the model.
If the cost of the model is $1 464, what is its height?
35. The sum of two numbers is 65.
(a) If one of the numbers is x, find the other number in terms of x.
(b) Given that the product of the two numbers is 816, find the two numbers.
36. The difference between two numbers is 11.
(a) If the smaller number is x, find the larger number in terms of x.
(b) Given that the product of the two numbers is 476, find the two numbers.
37. The largest number of three consecutive odd numbers is x and the product of the other two
numbers is 7x  8. What are the three numbers?
Ex.2F Advanced Set
n(n  1)
where n is a positive integer.
2
(a) If 1  2  3    n  351, find n.
(b) If 2  4  6    2n  650, find n.
38. It is given that 1  2  3    n 
39. In the figure, ABCD is a rectangular floor with length 9 m and width 6 m. PQRS is a
rectangular carpet put on the floor and leaves a uniform width x m of the floor uncovered
all round the room.
xm
D
C
S
xm
R
P
xm
Q
6m
A
9m
B
(a) Find the area of PQRS in terms of x.
(b) If the area of PQRS is one third the area of ABCD, find the dimensions of PQRS.
40. The sum of the areas of two squares is 89 cm 2 and the sum of their perimeters is 52 cm.
(a) If the side of one of the squares is x cm long, find the length of each side of another
square in terms of x.
(b) Hence find the lengths of each side of the two squares.
Chapter 2 Quadratic Equations
39
41. In the figure, ABCD is a trapezium. AB  2x cm, BC  5x cm, CD  (7x  1) cm and
AD  (x  2) cm.
A
(x  2) cm
D
2x cm B
5x cm
(7x  1) cm
C
(a) Find the values of x.
(b) Hence find the possible area of trapezium ABCD.
42. The product of two consecutive odd numbers is 2 499. Find the two numbers.
Level 2
28
. Find the two numbers.
195
44. Find the dimensions of a rectangle with a perimeter of 92 m and an a rea of
2 107 2
m .
4
45. Bruce is 32 years older than his son and 35 years older than his daughter. Three years later,
the product of the ages of Bruce’s children will be exactly the age of Bruce. How old is
Bruce now?
46. A man bought some bowls for $1 500 but twelve of them were broken. He sold each of the
remaining bowls at $6 more than the cost. Finally, he gained a profit of 69.2%. Find the
cost of each bowl.
47. Joan had bought a number of pens from a shop for $300 last week. The shop reduced the
price of each pen by $0.3 this week. If Joan bought the pens this week, she would buy 20
more pens by paying extra $8. How much was a pen after the price reduction?
48. A travel company charges $7 200 to a group of students for a trip. If 12 more students take
the trip, each student will pay $20 less to cover the charge. Find the original number of
students who planned to join the trip.
49. The distance between Kenny’s home and town A is 72 km. He drives from home to town A
at an average speed of x km/h and returns in the same route at an average speed of 3 km/h
less. The difference in time travelled between two trips is 1.2 hours. Find the value of x.
50. If the rate of the current is 3 km/h, Johnny spends 4.8 hours in rowing downstream o f 6 km
and returning upstream of 6 km. Find his rowing speed in still water.
51. Mary takes 5 days more than Susan to finish a piece of work. When they do it together,
they take 4 days less than Susan alone to finish the work. How long does Susan take t o
finish the work alone?
Ex.2F Advanced Set
43. The sum of the reciprocals of two consecutive odd numbers is
40
New Trend Mathematics S4A — Supplement
C HAPTER T EST
(Time allowed: 1 hour)
Section A
1. Solve the equation 4x 2  10 x  5 .
(3 marks)
2. Solve the equation (x  2)(x  6)  20.
(3 marks)
3. A root of the quadratic equation 5x 2  kx  36  0 is k.
(a) Find the values of k.
(2 marks)
(b) Using the result of (a), solve the equation 5x  kx  36  0 .
2
(2 marks)
4. Find the range of values of k if the quadratic equation kx2  3x  4  0 has real roots.
(4 marks)
5. (a) Find the value of k if the quadratic equation 5kx2  8kx  3k  1  0 has two equal real
roots.
(3 marks)
2
(b) Using the result of (a), solve the equation 5kx  8kx  3k  1  0 .
(2 marks)
6. In the figure, AF  (x  1) cm, AB  12 cm,
BC  14 cm and CD  (x  1) cm.
(a) Find EF in terms of x.
(1 mark)
(b) If the area of ABCDEF is 108 cm 2 , find
the length of DE.
(5 marks)
F
D
(x  1) cm
(x  1) cm
A
E
C
12 cm
14 cm
B
Section B
7. (a) The quadratic equation x 2  2(k  1) x  (k 2  5)  0 has a double root.
(i) Find the value of k.
(ii) Hence solve the equation x 2  2(k  1) x  (k 2  5)  0 .
(5 marks)
x 2
x
(b) (i) For the equation (
)  2(k  1)(
)  (k 2  5)  0 , if k is the value obtained in
x 1
x 1
(a)(i) , does the equation have a double root? Explain your answer.
x 2
x
(ii) Hence solve the equation (
(5 marks)
)  2(k  1)(
)  (k 2  5)  0 .
x 1
x 1
8. Lily takes 10 minutes to drive from her home to a supermarket at an average speed of
x
x km/h. Then she takes ( ) minutes more than the previous trip to drive from the
6
supermarket to a beach at an average speed of (x  22) km/h.
(a) Find the distance between Lily’s home and the supermarket in terms of x.
(2 marks)
(b) Find the distance between the supermarket and the beach in terms of x.
(3 marks)
(c) If the total distance travelled by Lily is 29 km, find t he time required for the whole trip.
(5 marks)
Chapter 2 Quadratic Equations
41
Multiple Choice Questions (3 marks each)
9. Which of the following equations has
1
roots 4 and  ?
4
2
A. 4 x  4 x  17  0
B. 4x 2  4x  17  0
C. 4 x  17 x  4  0
2
D. 4 x 2  17 x  4  0
10. If 4 is a root of 3( x  a ) 2  12 , find the
values of a.
A. 2 or 6
B. 2 or 4
C. 2 or 2
D. 2 or 4
11. If (x  2)(x  3)  (a  2)(a  3), then x 
A. a.
B. a or a  1.
C. a or 1  a.
D. a or a  1.
14. If x 
1
1
 4  , then x 
x
4
A. 4.
B. 
1
.
4
1
.
4
1
D. 4 or  .
4
C. 4 or
15. If x  2 x  3  0 , then x 
A.
B.
C.
D.
1.
3.
9.
1 or 9.
16. Solve the equation
( x  3) 2  6( x  3)  7  0 .
A.
B.
C.
D.
1 or 7
1 or 7
2 or 10
4 or 4
12. Find the value of the discriminant of the
equation 2x 2  5x  1 .
A. 33
B. 17
C. 16
D. 39
13. Which of the following equations has no
real roots?
I.
2 x  3x  5  0
2
II. 3x 2  5x  2  0
III. 5x 2  2 x  3  0
A.
B.
C.
D.
I and II only
I and III only
II and III only
I, II and III
17. The sum of two numbers is 20. If one of
the numbers is x and the product of the
two numbers is 99, which of the
following equations can be used to solve
the problem?
A. x 2  20 x  99  0
B. x 2  20 x  99  0
C. x 2  20 x  99  0
D. x 2  20 x  99  0
18. If the speed of a car increased by
10 km/h in a journey of 300 km, 1 hour
would have been saved. Find the original
speed of the car.
A. 40 km/h
B. 50 km/h
C. 60 km/h
D. 70 km/h
42
New Trend Mathematics S4A — Supplement
(for questions with
H INTS
in the textbook)
Revision Exercise 2
35. (b) Key information
The result obtained in (a).
Analysis
 A quadratic equation has real roots means that it has two equal or unequal real roots,
and therefore the value of the discriminant of the equation is greater than or equal to
zero.
 As the discriminant is expressed in terms of constants a and b and their values are
unknown, we can be sure that the value of the discriminant is always greater than or
equal to zero only if it can be written as a complete square.
Method
To factorize
the
discriminant
of
the
equation
by
applying
the
identity
x  2 xy  y  ( x  y ) .
2
2
2
36. (a) Key information
The quadratic equation ( x  a)( x  b)  c 2
Analysis
Since the equation is not written in general form, we cannot directly apply the formula
for finding the discriminant.
Method
Rewrite the quadratic equation into general form before applying the formula for
finding the discriminant. Then, expand (a  b) 2  4c 2 to compare with the expression
obtained for the discriminant.
38. (a) Key information
 Points B and E lie on the straight line y  kx.
 ABCD and DEFG are two squares each with one side lying on the x-axis.
 The coordinates of A are (2, 0).
Analysis
  Point D lies on the x-axis.
 The y-coordinate of D is zero.
  ABCD is a square.
 AB  AD
Thus knowing the length of AB can help find the x-coordinate of D.
Method
Since ABCD is a square, point B has the same x-coordinate as point A. Substitute x  2
into y  kx to find the y-coordinate of B and the length of AB can be found.