Quantitative Aptitude

Quadratic Equations

367 Questions
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Quadratic Equations Questions

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and mark the correct option. I. x2 + 6 + 7x = 0 II. y2 - 5y - 6 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or relation between x and y cannot be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

I: x^2 + 7x + 6 = 0 => (x+6)(x+1) = 0 => x = -6, -1. II: y^2 - 5y - 6 = 0 => (y-6)(y+1) = 0 => y = 6, -1. Comparing values: x can be -6 or -1, y can be 6 or -1. In all cases, x <= y.

Multiple choice

Directions: In the following question, two equations are given. You have to solve both the equations and choose the correct relationship accordingly. I. 7x2 + 17x - 12 = 0 II. 2y2 - 5y + 3 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

Solving 7x^2 + 17x - 12 = 0: (7x - 4)(x + 3) = 0, so x = 4/7 or -3. Solving 2y^2 - 5y + 3 = 0: (2y - 3)(y - 1) = 0, so y = 1.5 or 1. Comparing values, all possible x values are less than all possible y values.

Multiple choice

Directions: In the following question, two equations are given. You have to solve both the equations and choose the correct relationship accordingly. I. 27x2 + 30x + 8 = 0 II.3y2 + 5y + 2 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

I. 27x^2 + 30x + 8 = 0. Roots: x = (-30 +/- sqrt(900 - 864)) / 54 = (-30 +/- 6) / 54. x1 = -24/54 = -0.44, x2 = -36/54 = -0.66. II. 3y^2 + 5y + 2 = 0. (3y+2)(y+1) = 0. y1 = -2/3 = -0.66, y2 = -1. Comparing, x >= y.

Multiple choice

Solve x + y = x2 - y2 = 23 and find the value of y.

  1. 11

  2. 12

  3. 24

  4. 36
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

x + y = 23. x^2 - y^2 = (x+y)(x-y) = 23. Since x+y = 23, then 23(x-y) = 23, so x-y = 1. We have x+y=23 and x-y=1. Adding them: 2x = 24 => x=12. Subtracting: 2y = 22 => y=11.

Multiple choice

Directions: The following question consists of two equations. You are required to solve the equations and identify the correct answer from the alternatives given below. I. 3x2 – 4x + 1 = 0 II. y2 – 4y + 3 = 0

  1. x ≥ y

  2. x > y

  3. x ≤ y

  4. x < y

  5. x = y or the relation cannot be established.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Equation I: 3x^2 - 4x + 1 = 0 -> (3x-1)(x-1) = 0 -> x = 1/3, 1. Equation II: y^2 - 4y + 3 = 0 -> (y-3)(y-1) = 0 -> y = 1, 3. Comparing values: x=1/3 <= y=1, x=1/3 <= y=3, x=1 <= y=1, x=1 <= y=3. Thus x <= y.

Multiple choice

Directions: The following question consists of two equations. You are required to solve the equations and identify the correct answer from the alternatives given below. I. x2 – 20x + 75 = 0 II. 4y2 – 32y + 60 = 0

  1. x > y

  2. x ≥ y

  3. x ≤ y

  4. x < y

  5. x = y or the relation cannot be established.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

I: x^2 - 20x + 75 = 0 => (x-15)(x-5) = 0 => x = 5, 15. II: 4y^2 - 32y + 60 = 0 => y^2 - 8y + 15 = 0 => (y-5)(y-3) = 0 => y = 3, 5. Comparing values: x can be 5 or 15, y can be 3 or 5. In all cases, x >= y.

Multiple choice

Directions: There are two equations given in the problem. Solve the equations and answer accordingly. I. x2 - 22x + 117 = 0 II. 2y2 + y - 120 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

  5. x = y or no relation can be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For x^2 - 22x + 117 = 0, the roots are (x-9)(x-13)=0, so x = 9, 13. For 2y^2 + y - 120 = 0, using the quadratic formula, y = (-1 +/- sqrt(1 + 960)) / 4 = (-1 +/- 31) / 4, giving y = 7.5 or -8. Comparing the sets {9, 13} and {7.5, -8}, all values of x are greater than all values of y.

Multiple choice

Directions: In the following question, equations (i) and (ii) are given. You have to solve both the equations and give answer accordingly. (i) 15x2 + 17x + 4 = 0 (ii) y2 + 18y + 65 = 0

  1. x ≥ y

  2. x < y

  3. x ≤ y

  4. x > y

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

For (i) 15x^2 + 17x + 4 = 0, roots are x = -0.5, -0.66. For (ii) y^2 + 18y + 65 = 0, roots are y = -5, -13. Since all values of x are greater than all values of y, x > y.

Multiple choice

Directions: In the following question, equations (i) and (ii) are given. You have to solve both the equations and give answer accordingly. (i) 56x2 + 30x + 4 = 0 (ii) 64y2 + 28y + 3 = 0

  1. x ≥ y

  2. x < y

  3. x ≤ y

  4. x > y

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

For 56x^2 + 30x + 4 = 0, roots are negative. For 64y^2 + 28y + 3 = 0, roots are negative. Solving for x: x = (-30 +/- sqrt(900 - 896))/112 = (-30 +/- 2)/112. Roots are -28/112 = -0.25 and -32/112 = -0.285. For y: y = (-28 +/- sqrt(784 - 768))/128 = (-28 +/- 4)/128. Roots are -24/128 = -0.1875 and -32/128 = -0.25. Comparing values, x <= y.

Multiple choice

Directions: In the following question, two equations numbered (I) and (II) are given. Student should solve both the equations and mark appropriate answer. I. 12x2 + 15x + 3 = 0 II. 14y2 + 15y + 4 = 0

  1. If x = y or no relation can be established

  2. If x > y

  3. If x < y

  4. If x ≥ y

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

Equation I: 12x^2 + 15x + 3 = 0 -> 4x^2 + 5x + 1 = 0 -> (4x+1)(x+1) = 0. Roots are -0.25, -1. Equation II: 14y^2 + 15y + 4 = 0 -> (2y+1)(7y+4) = 0. Roots are -0.5, -0.57. Comparing roots: -0.25 > -0.5 and -0.25 > -0.57, but -1 < -0.5. No relation can be established.

Multiple choice

Directions: In the following question, two equations numbered (I) and (II) are given. Student should solve both the equations and mark appropriate answer. I. x3 = 729 II. 3y2 - 20y + 28 = 0

  1. If x = y or no relation can be established

  2. If x > y

  3. If x < y

  4. If x ≥ y

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

Solving equation (I), x^3 = 729 gives x = 9. Solving equation (II), 3y^2 - 20y + 28 = 0, we factor to (3y - 14)(y - 2) = 0, giving y = 14/3 (approx 4.67) or y = 2. Since 9 > 4.67 and 9 > 2, x > y holds.

Multiple choice

Directions: In the following question, two equations numbered (I) and (II) are given. Student should solve both the equations and mark appropriate answer. I. 2x2 - 12x + 18 = 0 II. y2 + 18 = 459

  1. If x < y

  2. If x > y

  3. If x = y or no relation can be established

  4. If x ≥ y

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

I: 2x^2 - 12x + 18 = 0 => x^2 - 6x + 9 = 0 => (x-3)^2 = 0 => x = 3. II: y^2 + 18 = 459 => y^2 = 441 => y = 21 or -21. Comparing x=3 with y=21 and y=-21, we see x < y (if y=21) and x > y (if y=-21). Thus, no relation can be established.

Multiple choice

Directions: In the question, two equations numbered (I) and (II) are given. You have to solve both the equations and then mark the appropriate answer. (I) 8x2 – 144x + 648 = 0 (II) y2 – 5y – 36 = 0

  1. x = y or no relationship can be established between them.

  2. x > y

  3. x < y

  4. x ≥ y

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

Eq I: 8x^2 - 144x + 648 = 0 -> x^2 - 18x + 81 = 0 -> (x-9)^2 = 0 -> x = 9. Eq II: y^2 - 5y - 36 = 0 -> (y-9)(y+4) = 0 -> y = 9, -4. Comparing x=9 with y=9 or -4, x >= y.

Multiple choice

Directions: In the following question, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 4x2 + 11x + 7 = 0 (ii) 3y2 + 15y + 18 = 0

  1. x ≥ y

  2. x > y

  3. x ≤ y

  4. x < y

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

For (i) 4x^2 + 11x + 7 = 0, roots are x = -1 and x = -7/4 (-1.75). For (ii) 3y^2 + 15y + 18 = 0, y^2 + 5y + 6 = 0, roots are y = -2 and y = -3. Comparing the roots, -1 > -2, -1 > -3, -1.75 > -2, and -1.75 > -3. Thus, x > y.