87 Quadratic Equation. 100 Important Quadratic Equations Practice Questions for SSC Class 10

Part 1: Top 10 Solved Quadratic Equations

Q1. Solve by factorization: \(x^2 + 5x + 6 = 0\)

Step 1: Find two numbers whose sum is 5 and product is 6. The numbers are 2 and 3. Step 2: \(x^2 + 2x + 3x + 6 = 0\) Step 3: \(x(x + 2) + 3(x + 2) = 0\) Step 4: \((x + 2)(x + 3) = 0\) Step 5: \(x + 2 = 0\) or \(x + 3 = 0\) Final Answer: \(x = -2, x = -3\)

Q2. Solve by factorization: \(x^2 - 3x - 10 = 0\)

Sum = -3, Product = -10. Numbers are -5 and 2. \(x^2 - 5x + 2x - 10 = 0\) \(x(x - 5) + 2(x - 5) = 0\) \((x - 5)(x + 2) = 0\) Final Answer: \(x = 5, x = -2\)

Q3. Solve using Formula Method: \(x^2 + 6x + 5 = 0\)

Comparing with \(ax^2 + bx + c = 0\), \(a=1, b=6, c=5\) Discriminant \((\Delta) = b^2 - 4ac = 6^2 - 4(1)(5) = 36 - 20 = 16\) Using formula: \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) \(x = \frac{-6 \pm \sqrt{16}}{2} = \frac{-6 \pm 4}{2}\) \(x = \frac{-2}{2} = -1\) or \(x = \frac{-10}{2} = -5\) Final Answer: \(x = -1, -5\)

Q4. Solve: \(2y^2 + 27y + 13 = 0\)

Sum = 27, Product = \(2 \times 13 = 26\). Numbers: 26, 1. \(2y^2 + 26y + 1y + 13 = 0\) \(2y(y + 13) + 1(y + 13) = 0\) \((y + 13)(2y + 1) = 0\) Final Answer: \(y = -13, y = -1/2\)

Q5. Find the value of Discriminant for \(x^2 + 7x - 1 = 0\)

\(a = 1, b = 7, c = -1\) \(\Delta = b^2 - 4ac\) \(\Delta = (7)^2 - 4(1)(-1) = 49 + 4 = 53\) Final Answer: \(\Delta = 53\)

Q6. Determine the nature of roots for \(x^2 - 4x + 4 = 0\)

\(\Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0\) Since \(\Delta = 0\), the roots are real and equal. Final Answer: Real and Equal roots.

Q7. Form a quadratic equation whose roots are 3 and 10.

Let \(\alpha = 3, \beta = 10\) \(\alpha + \beta = 13\), \(\alpha\beta = 30\) Equation: \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\) Final Answer: \(x^2 - 13x + 30 = 0\)

Q8. Find \(k\) if \(x = 3\) is a root of \(kx^2 - 10x + 3 = 0\)

Substitute \(x = 3\) in the equation: \(k(3)^2 - 10(3) + 3 = 0\) \(9k - 30 + 3 = 0 \implies 9k - 27 = 0\) \(9k = 27 \implies k = 3\) Final Answer: \(k = 3\)

Q9. Solve: \(x^2 - 15x + 54 = 0\)

Numbers: -9, -6 (Sum -15, Product 54) \((x - 9)(x - 6) = 0\) Final Answer: \(x = 9, 6\)

Q10. Solve by completing the square: \(x^2 + x - 20 = 0\)

\(x^2 + x = 20\) Add \((\frac{1}{2})^2 = \frac{1}{4}\) to both sides. \(x^2 + x + \frac{1}{4} = 20 + \frac{1}{4} \implies (x + \frac{1}{2})^2 = \frac{81}{4}\) \(x + \frac{1}{2} = \pm \frac{9}{2}\) \(x = \frac{8}{2} = 4\) or \(x = \frac{-10}{2} = -5\) Final Answer: \(x = 4, -5\)

Part 2: Practice Questions (Q11 - Q100)

Solve the following Quadratic Equations (Factorization/Formula):

11. \(x^2 - 4x - 5 = 0\)
12. \(x^2 + 8x + 15 = 0\)
13. \(x^2 - 7x + 12 = 0\)
14. \(2x^2 - 5x + 2 = 0\)
15. \(3x^2 - x - 10 = 0\)
16. \(x^2 - 11x + 24 = 0\)
17. \(x^2 + 2x - 48 = 0\)
18. \(5m^2 = 22m + 15\)
19. \(2x^2 - 2x + \frac{1}{2} = 0\)
20. \(6x - \frac{2}{x} = 1\)
21. \(x^2 - 25 = 0\)
22. \(3y^2 = 15y\)
23. \(x^2 + 4x + 1 = 0\)
24. \(m^2 - 5m - 3 = 0\)
25. \(x^2 + 5x + 5 = 0\)
26. \(y^2 + \frac{1}{3}y = 2\)
27. \(5x^2 + 13x + 8 = 0\)
28. \(x^2 + 10x + 24 = 0\)
29. \(x^2 - x - 72 = 0\)
30. \(x^2 - 16x + 63 = 0\)
31. \(2x^2 + 9x + 10 = 0\)
32. \(x^2 - 2x - 3 = 0\)
33. \(x^2 + 6x + 9 = 0\)
34. \(x^2 - 10x + 25 = 0\)
35. \(x^2 + 14x + 49 = 0\)
36. \(4x^2 - 4x + 1 = 0\)
37. \(x^2 - 1 = 0\)
38. \(2x^2 - 7x + 6 = 0\)
39. \(3x^2 + 8x + 5 = 0\)
40. \(x^2 - 12x + 32 = 0\)

Find the Discriminant and State Nature of Roots:

41. \(x^2 + x + 1 = 0\)
42. \(2x^2 - 5x - 3 = 0\)
43. \(x^2 - 6x + 9 = 0\)
44. \(3x^2 + 2x - 1 = 0\)
45. \(x^2 + 4x + 5 = 0\)
46. \(2x^2 - 7x + 3 = 0\)
47. \(x^2 - 2x + 1 = 0\)
48. \(x^2 + 5x + 6 = 0\)
49. \(4x^2 - 12x + 9 = 0\)
50. \(x^2 - 8x + 15 = 0\)
51. \(2x^2 + 5x + 5 = 0\)
52. \(x^2 - x - 1 = 0\)
53. \(x^2 + 10x + 25 = 0\)
54. \(3x^2 + 7x + 2 = 0\)
55. \(x^2 + 2x + 3 = 0\)
56. \(5x^2 - 4x + 1 = 0\)
57. \(x^2 - 4x + 3 = 0\)
58. \(2x^2 - 6x + 3 = 0\)
59. \(x^2 + 12x + 36 = 0\)
60. \(x^2 - 5x + 7 = 0\)

Form Quadratic Equations from Roots:

61. Roots: 2, 5
62. Roots: -3, -4
63. Roots: 0, 4
64. Roots: 1/2, 1/2
65. Roots: \(\sqrt{2}, -\sqrt{2}\)
66. Roots: 7, -7
67. Roots: 6, 1
68. Roots: -5, 2
69. Roots: 8, 3
70. Roots: -1, -1
71. Roots: 10, -2
72. Roots: 0, 0
73. Roots: 5, 5
74. Roots: -6, 6
75. Roots: 4, -3
76. Roots: 1, 9
77. Roots: -2, -8
78. Roots: 3, -3
79. Roots: 1/3, 3
80. Roots: -10, -10

Advanced and Word-Based Conditions:

81. Find \(k\) if roots of \(x^2 + kx + 12 = 0\) are real and equal.
82. Find \(k\) if one root of \(x^2 - kx + 18 = 0\) is 6.
83. Sum of roots is 10 and product is 21. Find equation.
84. One root is \(2 + \sqrt{3}\), find the other root.
85. Solve \(x^4 - 5x^2 + 4 = 0\) (Reducible to quadratic).
86. Solve \((x-3)(x+4) = 0\).
87. Solve \(x^2 = 49\).
88. Solve \(5x^2 = 20\).
89. Find \(k\) if \(\Delta = 0\) for \(kx(x-2) + 6 = 0\).
90. Solve \(x + 1/x = 2.5\).
91. Product of two consecutive natural numbers is 20.
92. Find \(x\) if \(x^2 - 9x + 20 = 0\).
93. Solve \(y^2 + 10y + 21 = 0\).
94. Solve \(x^2 - 11x + 30 = 0\).
95. Solve \(x^2 - 2x - 8 = 0\).
96. Roots are 1 and -1. Form equation.
97. Roots are 4 and 0. Form equation.
98. Find \(\Delta\) for \(x^2 + 5x + 5 = 0\).
99. Solve \(x^2 - 3x = 0\).
100. Solve \(2x^2 = 8\).

Answer Key (Q11 - Q100)

11. 5, -1

12. -3, -5

13. 4, 3

14. 2, 0.5

15. 2, -5/3

16. 8, 3

17. 6, -8

18. 5, -3/5

19. 0.5, 0.5

20. 2/3, -1/2

21. 5, -5

22. 0, 5

23. \(-2 \pm \sqrt{3}\)

24. \(\frac{5 \pm \sqrt{37}}{2}\)

25. \(\frac{-5 \pm \sqrt{5}}{2}\)

26. 4/3, -3/2

27. -1, -1.6

28. -4, -6

29. 9, -8

30. 9, 7

31. -2, -2.5

32. 3, -1

33. -3, -3

34. 5, 5

35. -7, -7

36. 0.5, 0.5

37. 1, -1

38. 2, 1.5

39. -1, -5/3

40. 8, 4

41. -3 (Not Real)

42. 49 (Real, Uniq)

43. 0 (Real, Equal)

44. 16 (Real, Uniq)

45. -4 (Not Real)

46. 25 (Real, Uniq)

47. 0 (Real, Equal)

48. 1 (Real, Uniq)

49. 0 (Real, Equal)

50. 4 (Real, Uniq)

51. -15 (Not Real)

52. 5 (Real, Uniq)

53. 0 (Real, Equal)

54. 25 (Real, Uniq)

55. -8 (Not Real)

56. -4 (Not Real)

57. 4 (Real, Uniq)

58. 12 (Real, Uniq)

59. 0 (Real, Equal)

60. -3 (Not Real)

61. \(x^2-7x+10=0\)

62. \(x^2+7x+12=0\)

63. \(x^2-4x=0\)

64. \(4x^2-4x+1=0\)

65. \(x^2-2=0\)

66. \(x^2-49=0\)

67. \(x^2-7x+6=0\)

68. \(x^2+3x-10=0\)

69. \(x^2-11x+24=0\)

70. \(x^2+2x+1=0\)

71. \(x^2-8x-20=0\)

72. \(x^2=0\)

73. \(x^2-10x+25=0\)

74. \(x^2-36=0\)

75. \(x^2-x-12=0\)

76. \(x^2-10x+9=0\)

77. \(x^2+10x+16=0\)

78. \(x^2-9=0\)

79. \(3x^2-10x+3=0\)

80. \(x^2+20x+100=0\)

81. \(k = \pm \sqrt{48}\)

82. \(k = 9\)

83. \(x^2-10x+21=0\)

84. \(2-\sqrt{3}\)

85. \(\pm 1, \pm 2\)

86. 3, -4

87. 7, -7

88. 2, -2

89. \(k = 6\)

90. 2, 0.5

91. 4, 5

92. 4, 5

93. -3, -7

94. 6, 5

95. 4, -2

96. \(x^2-1=0\)

97. \(x^2-4x=0\)

98. 5

99. 0, 3

100. 2, -2