10 Quadratic Equation Questions Important for Board Exam

Part 1: 10 Important Solved Questions

SSC Class 10 Algebra Quadratic Equations Important Solved Questions and Practice Set

Question 1: Factorization Method

Solve the quadratic equation by factorization: \(x^2 - 15x + 54 = 0\)

Solution:
Given: \(x^2 - 15x + 54 = 0\)
We need to find two numbers whose sum is \(-15\) and product is \(54\).
The numbers are \(-9\) and \(-6\).
\(x^2 - 9x - 6x + 54 = 0\)
\(x(x - 9) - 6(x - 9) = 0\)
\((x - 9)(x - 6) = 0\)
\(x - 9 = 0\) or \(x - 6 = 0\)
\(x = 9\) or \(x = 6\)
Roots: 9, 6

Question 2: Formula Method

Solve using formula: \(x^2 + 6x + 5 = 0\)

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

Question 3: Nature of Roots

Determine the nature of roots for \(2x^2 - 5x + 7 = 0\)

Solution:
\(a = 2, b = -5, c = 7\)
\(\Delta = b^2 - 4ac = (-5)^2 - 4(2)(7) = 25 - 56 = -31\)
Since \(\Delta < 0\), the roots are not real.

Question 4: Finding 'k'

Find \(k\) if the roots of \(kx(x - 2) + 6 = 0\) are real and equal.

Solution:
Equation: \(kx^2 - 2kx + 6 = 0\)
For real and equal roots, \(\Delta = 0\).
\(a = k, b = -2k, c = 6\)
\((-2k)^2 - 4(k)(6) = 0\)
\(4k^2 - 24k = 0\)
\(4k(k - 6) = 0\)
Since \(k \neq 0\) (as it is a quadratic equation), \(k - 6 = 0 \implies \mathbf{k = 6}\).

Question 5: Forming Equation

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

Solution:
Let \(\alpha = 3\) and \(\beta = -10\).
Sum of roots \(\alpha + \beta = 3 + (-10) = -7\)
Product of roots \(\alpha\beta = 3 \times (-10) = -30\)
Equation: \(x^2 - (\alpha + \beta)x + \alpha\beta = 0\)
\(x^2 - (-7)x + (-30) = 0\)
Equation: \(x^2 + 7x - 30 = 0\)

Question 6: Sum and Product Relation

If \(\alpha\) and \(\beta\) are roots of \(x^2 + 5x - 1 = 0\), find \(\alpha^3 + \beta^3\).

Solution:
\(\alpha + \beta = -b/a = -5\), \(\alpha\beta = c/a = -1\)
\(\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)\)
\(= (-5)^3 - 3(-1)(-5)\)
\(= -125 - 15 = -140\)
Value: -140

Question 7: Word Problem (Numbers)

Product of Pragati's age 2 years ago and 3 years hence is 84. Find her present age.

Solution:
Let present age be \(x\).
\((x - 2)(x + 3) = 84\)
\(x^2 + 3x - 2x - 6 = 84\)
\(x^2 + x - 90 = 0\)
\((x + 10)(x - 9) = 0\)
\(x = -10\) (Discarded) or \(x = 9\).
Present age: 9 years

Question 8: Completing the Square

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

Solution:
\(x^2 + x = 20\)
Add \((1/2 \times \text{coeff of } x)^2 = (1/2)^2 = 1/4\) to both sides:
\(x^2 + x + 1/4 = 20 + 1/4\)
\((x + 1/2)^2 = 81/4\)
Taking square root: \(x + 1/2 = \pm 9/2\)
\(x = 9/2 - 1/2 = 4\) or \(x = -9/2 - 1/2 = -5\)
Roots: 4, -5

Question 9: Alpha/Beta Expression

If roots of \(x^2 - px + q = 0\) differ by 1, prove \(p^2 = 4q + 1\).

Solution:
Let roots be \(\alpha, \beta\). Given \(|\alpha - \beta| = 1\).
\(\alpha + \beta = p, \alpha\beta = q\).
We know \((\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\)
\(1^2 = p^2 - 4q\)
\(1 = p^2 - 4q \implies \mathbf{p^2 = 4q + 1}\). (Hence Proved)

Question 10: Speed/Distance

A train travels 360 km at uniform speed. If speed was 5 km/hr more, it would take 1 hr less. Find the speed.

Solution:
Let original speed be \(x\) km/hr. Time \(T_1 = 360/x\).
New speed \(x + 5\). Time \(T_2 = 360/(x + 5)\).
\(T_1 - T_2 = 1\)
\(360/x - 360/(x + 5) = 1\)
\(360(x + 5 - x) = x(x + 5)\)
\(1800 = x^2 + 5x \implies x^2 + 5x - 1800 = 0\)
\((x + 45)(x - 40) = 0\)
Speed cannot be negative, so \(x = 40\).
Speed: 40 km/hr

Part 2: 50 Practice Questions

1. \(x^2 - 4x + 3 = 0\)

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

3. \(x^2 - 5x + 6 = 0\)

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

5. \(2x^2 - 7x + 3 = 0\)

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

7. \(3x^2 - 5x + 2 = 0\)

8. \(x^2 + 2x - 8 = 0\)

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

10. \(x^2 - 6x + 8 = 0\)

11. \(x^2 + 5x + 4 = 0\)

12. \(2x^2 + x - 6 = 0\)

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

14. \(x^2 - 11x + 30 = 0\)

15. \(x^2 + 8x + 15 = 0\)

16. \(x^2 - 13x + 40 = 0\)

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

18. \(5x^2 - 18x - 8 = 0\)

19. \(x^2 + 4x + 4 = 0\)

20. \(x^2 - 12x + 32 = 0\)

21. \(2x^2 - 5x - 3 = 0\)

22. \(x^2 - 2x - 15 = 0\)

23. \(x^2 + 9x + 20 = 0\)

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

25. \(3x^2 - 10x + 3 = 0\)

26. \(x^2 - 14x + 49 = 0\)

27. \(x^2 - 16 = 0\)

28. \(2x^2 + 7x + 5 = 0\)

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

30. \(x^2 - 15x + 56 = 0\)

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

32. \(x^2 - 8x + 12 = 0\)

33. \(4x^2 - 4x + 1 = 0\)

34. \(x^2 - 25 = 0\)

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

36. \(x^2 - 5x - 14 = 0\)

37. \(x^2 - 11x + 28 = 0\)

38. \(x^2 + 6x + 9 = 0\)

39. \(2x^2 - 3x + 1 = 0\)

40. \(x^2 - 100 = 0\)

41. \(x^2 - 17x + 72 = 0\)

42. \(x^2 + 12x + 35 = 0\)

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

44. \(3x^2 + 4x + 1 = 0\)

45. \(x^2 - 9x + 20 = 0\)

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

47. \(x^2 + 2x - 15 = 0\)

48. \(x^2 - 13x + 42 = 0\)

49. \(x^2 - 64 = 0\)

50. \(2x^2 - x - 1 = 0\)

🔑 Answer Key

1: (3, 1)
2: (-2, -5)
3: (2, 3)
4: (3, -3)
5: (3, 0.5)
6: (5, 5)
7: (1, 2/3)
8: (2, -4)
9: (1, -1)
10: (4, 2)
11: (-1, -4)
12: (1.5, -2)
13: (5, -2)
14: (5, 6)
15: (-3, -5)
16: (5, 8)
17: (2, -2)
18: (4, -0.4)
19: (-2, -2)
20: (4, 8)
21: (3, -0.5)
22: (5, -3)
23: (-4, -5)
24: (3, 4)
25: (3, 1/3)
26: (7, 7)
27: (4, -4)
28: (-1, -2.5)
29: (4, -3)
30: (7, 8)
31: (-3, -7)
32: (2, 6)
33: (0.5, 0.5)
34: (5, -5)
35: (2, -3)
36: (7, -2)
37: (4, 7)
38: (-3, -3)
39: (1, 0.5)
40: (10, -10)
41: (8, 9)
42: (-5, -7)
43: (1, 1)
44: (-1/3, -1)
45: (4, 5)
46: (6, -6)
47: (3, -5)
48: (6, 7)
49: (8, -8)
50: (1, -0.5)