Quantitative Aptitude

Quadratic Equations

367 Questions
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Quadratic Equations Questions

Multiple choice

Directions: In the given question, two quantities are given, one as Quantity I and another as Quantity II. You have to determine the relationship between the two quantities and choose the appropriate option. 4x2 - 23x + 33 = 0 Quantity I: x Quantity II: 4

  1. Quantity I > Quantity II

  2. Quantity I ≤ Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I < Quantity II

  5. Quantity I = Quantity II, or the relationship cannot be established from the information given.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solving 4x^2 - 23x + 33 = 0 using the quadratic formula or factoring: (4x - 11)(x - 3) = 0. The roots are x = 11/4 = 2.75 and x = 3. Since both values are less than 4, Quantity I < Quantity II.

Multiple choice

Directions: In the given question, two quantities are given, one as Quantity I and another as Quantity II. You have to determine the relationship between the two quantities and choose the appropriate option. 20x2 + 82x + 78 = 0 Quantity I: x2 Quantity II: (x - 1)

  1. Quantity I > Quantity II

  2. Quantity I ≤ Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I < Quantity II

  5. Quantity I = Quantity II, or the relationship cannot be established from the information given.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

20x^2 + 82x + 78 = 0. Divide by 2: 10x^2 + 41x + 39 = 0. Factors: (2x + 3)(5x + 13) = 0. Roots: x = -1.5 or x = -2.6. If x = -1.5, Q1 = 2.25, Q2 = -2.5 (Q1 > Q2). If x = -2.6, Q1 = 6.76, Q2 = -3.6 (Q1 > Q2). In both cases, Q1 > Q2.

Multiple choice

Directions: In the given question, two quantities are given, one as Quantity I and another as Quantity II. You have to determine the relationship between the two quantities and choose the appropriate option. 2x2 + 27x - 45 = 0 Quantity I: x + 7 Quantity II: x2 + 7

  1. Quantity I > Quantity II

  2. Quantity I ≤ Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I < Quantity II

  5. Quantity I = Quantity II, or the relationship cannot be established from the information given.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solving 2x^2 + 27x - 45 = 0: (2x - 3)(x + 15) = 0. Roots are x = 1.5 or x = -15. If x = 1.5, Q1 = 8.5, Q2 = 9.25. If x = -15, Q1 = -8, Q2 = 232. In both cases, Q1 < Q2.

Multiple choice

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 11x + 64/x = 54 II. 12y2 + 40y + 17 = 0

  1. x > y

  2. x ≤ y

  3. x ≥ y

  4. x < y

  5. x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

I: 11x + 64/x = 54 => 11x^2 - 54x + 64 = 0. Roots are x = 4 and x = 16/11 (approx 1.45). II: 12y^2 + 40y + 17 = 0. Roots are y = -0.5 and y = -2.83. Since both roots of x are positive and both roots of y are negative, x > y.

Multiple choice

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 2x2 + 5x + 2 = 0 II. 6y2 + 69y + 198 = 0

  1. x > y

  2. x ≤ y

  3. x ≥ y

  4. x < y

  5. Either x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Eq I: 2x^2 + 5x + 2 = 0 => (2x+1)(x+2) = 0 => x = -0.5, -2. Eq II: 6y^2 + 69y + 198 = 0 => 2y^2 + 23y + 66 = 0 => (2y+11)(y+6) = 0 => y = -5.5, -6. Comparing values, -0.5 > -5.5 and -0.5 > -6, while -2 > -5.5 and -2 > -6. Thus, x > y.

Multiple choice

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 20x2 – 108x + 144 = 0 II. 8y2 + 18y + 4 = 0

  1. x > y

  2. x ≤ y

  3. x ≥ y

  4. x < y

  5. Either x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

I: 20x^2 - 108x + 144 = 0 -> 5x^2 - 27x + 36 = 0. Roots are (27 +/- sqrt(729 - 720))/10 = (27 +/- 3)/10, so x = 3 or 2.4. II: 8y^2 + 18y + 4 = 0 -> 4y^2 + 9y + 2 = 0. Roots are (-9 +/- sqrt(81 - 32))/8 = (-9 +/- 7)/8, so y = -0.25 or -2. Comparing x and y, x > y.

Multiple choice

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 3x2 – 5x – 12 = 0 II. 2y2 + 15y + 25 = 0

  1. x > y

  2. x ≤ y

  3. x ≥ y

  4. x < y

  5. Either x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

I: 3x^2 - 5x - 12 = 0 => (3x+4)(x-3) = 0 => x = -4/3, 3. II: 2y^2 + 15y + 25 = 0 => (2y+5)(y+5) = 0 => y = -2.5, -5. Comparing values, x is always greater than y.

Multiple choice

Directions: In this question, two equations numbered I and II are given. You have to solve both the equations and mark the correct answer. (I) x + 2 = x2 (II) 35y + 2 = 37y2

  1. x > y

  2. x < y

  3. x ≥ y

  4. x = y or no relationship can be established between 'x' and 'y'

  5. x ≤ y
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Equation I: x^2 - x - 2 = 0 => (x-2)(x+1) = 0, so x = 2 or -1. Equation II: 37y^2 - 35y - 2 = 0 => (37y + 2)(y - 1) = 0, so y = 1 or -2/37. Comparing the sets {2, -1} and {1, -0.054}, no consistent relationship exists (e.g., 2 > 1 but -1 < 1).

Multiple choice

Directions: In this question, two equations numbered I and II are given. You have to solve both the equations and mark the correct answer. (I) 14x2 - 51x = -7 (II) 1 - 6y - 7y2 = 0

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

  5. x = y or no relationship can be established between 'x' and 'y'
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

I: 14x^2 - 51x + 7 = 0. Roots are (51 +/- sqrt(2601 - 392))/28 = (51 +/- 47)/28. x = 98/28 = 3.5 or 4/28 = 1/7. II: 7y^2 + 6y - 1 = 0. (7y-1)(y+1) = 0. y = 1/7 or -1. Comparing x and y, x >= y.

Multiple choice

Directions: In this question, two equations numbered I and II are given. You have to solve both the equations and mark the correct answer. (I) 2x2 = -3x - 1 (II) 61y2 = 123y - 2

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

  5. x = y or no relationship can be established between 'x' and 'y'
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Equation I: 2x^2 + 3x + 1 = 0 factors to (2x + 1)(x + 1) = 0, so x = -0.5 or -1. Equation II: 61y^2 - 123y + 2 = 0. Using the quadratic formula, y = (123 +/- sqrt(123^2 - 4 * 61 * 2)) / (2 * 61) = (123 +/- sqrt(15129 - 488)) / 122 = (123 +/- sqrt(14641)) / 122 = (123 +/- 121) / 122. Thus y = 244/122 = 2 or y = 2/122 = 1/61. Comparing values, x is always less than y.

Multiple choice

Directions: In this question, two equations numbered I and II are given. You have to solve both the equations and mark the correct answer. (I) 2x2 - 1 + x = 0 (II) 7y + 2 = 22y2

  1. x > y

  2. x < y

  3. x = y or no relationship can be established between 'x' and 'y'

  4. x ≤ y

  5. x ≥ y
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Equation I: 2x^2 + x - 1 = 0 factors to (2x - 1)(x + 1) = 0, so x = 0.5 or -1. Equation II: 22y^2 - 7y - 2 = 0 factors to (11y + 2)(2y - 1) = 0, so y = -2/11 or 0.5. Comparing these values, x can be greater than, less than, or equal to y depending on the specific roots chosen, so no unique relationship exists.

Multiple choice

Directions: In this question, two equations numbered I and II are given. You have to solve both the equations and mark the correct answer. (I) 7x = 5 + 2x2 (II) 53y - 22y2 + 5 = 0

  1. x > y

  2. x = y or no relationship can be established between 'x' and 'y'

  3. x ≥ y

  4. x ≤ y

  5. x < y
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

I: 2x^2 - 7x + 5 = 0 => (2x-5)(x-1) = 0 => x = 2.5, 1. II: 22y^2 - 53y - 5 = 0. Roots using quadratic formula: (53 +/- sqrt(2809 + 440))/44 = (53 +/- 57)/44. y = 110/44 = 2.5 or y = -4/44 = -0.09. Since x can be 2.5 and y can be 2.5, or x can be 1 and y can be -0.09, no relationship is established.

Multiple choice

Directions: In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. 25x2 – 90x + 72 = 0II. y2 + 26y + 168 = 0

  1. x > y

  2. x ≤ y

  3. x ≥ y

  4. x < y

  5. x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Eq I: 25x^2 - 90x + 72 = 0. Roots: x = [90 +/- sqrt(8100 - 7200)] / 50 = (90 +/- 30) / 50. x = 1.2 or 2.4. Eq II: y^2 + 26y + 168 = 0. Roots: y = [-26 +/- sqrt(676 - 672)] / 2 = (-26 +/- 2) / 2. y = -12 or -14. Since 1.2 and 2.4 are both greater than -12 and -14, x > y.

Multiple choice

Directions: In the given question, two equations are given. You have to solve both the equations and mark the correct option. (1) x2 - 5.2x + 6.51 = 0 (2) y2 - 9.2y + 20.91 = 0

  1. x > y

  2. x < y

  3. x ≤ y

  4. x ≥ y

  5. The relationship between x and y cannot be established.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

(1) x^2 - 5.2x + 6.51 = 0. Roots: x = [5.2 +/- sqrt(27.04 - 26.04)] / 2 = [5.2 +/- 1] / 2. Roots: 3.1, 2.1. (2) y^2 - 9.2y + 20.91 = 0. Roots: y = [9.2 +/- sqrt(84.64 - 83.64)] / 2 = [9.2 +/- 1] / 2. Roots: 5.1, 4.1. Comparing all pairs: 3.1 < 5.1, 3.1 < 4.1, 2.1 < 5.1, 2.1 < 4.1. Thus x < y.