Quantitative Aptitude

Quadratic Equations

367 Questions
All Basic factorization 305 Sign scheme based 22 Algebraic substitution 19 Roots from relation 9 Finding extrema 5 Discriminant analysis 2 Finding coefficients 2 Common root 1 Finding quadratic equation 1 Reciprocal equation 1

Quadratic Equations Questions

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) 7x2 + 17x + 10 = 0 (ii) 9y2 + 15y + 6 = 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
C Correct answer
Explanation

i) 7x^2 + 17x + 10 = 0 => (7x+10)(x+1) = 0 => x = -10/7, -1. ii) 9y^2 + 15y + 6 = 0 => 3(3y^2 + 5y + 2) = 0 => 3(3y+2)(y+1) = 0 => y = -2/3, -1. Comparing: -1.428 <= -0.66 and -1 <= -0.66. So 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) 7x2 – 15x + 2 = 0 (ii) 8y2 – 10y + 3 = 0

  1. x = y or no relation can be established between x & y.

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

Solving 7x^2 - 15x + 2 = 0: (7x - 1)(x - 2) = 0, so x = 1/7 or 2. Solving 8y^2 - 10y + 3 = 0: (4y - 3)(2y - 1) = 0, so y = 3/4 or 1/2. Comparing values: 2 > 3/4, 2 > 1/2, 1/7 < 3/4, 1/7 < 1/2. Since x can be greater or smaller than y, no relation can be established.

Multiple choice

Directions: In this question, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 12x2 + 11x + 2 = 0 (ii) 3y2 + 20y + 32 = 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
D Correct answer
Explanation

For (i) 12x^2 + 11x + 2 = 0, roots are x = -1/4, -2/3. For (ii) 3y^2 + 20y + 32 = 0, roots are y = -4, -8/3. Comparing values: -0.25 > -4, -0.25 > -2.66, -0.66 > -4, -0.66 > -2.66. Thus, x > y.

Multiple choice

Directions: In this question, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 4x2 – 12x + 9 = 0 (ii) 16y2 – 24y + 5 = 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
D Correct answer
Explanation

Solving 4x^2 - 12x + 9 = 0 gives (2x - 3)^2 = 0, so x = 1.5. Solving 16y^2 - 24y + 5 = 0 using the quadratic formula gives y = (24 +/- sqrt(576 - 320))/32 = (24 +/- 16)/32, resulting in y = 1.25 or y = 0.25. Since 1.5 > 1.25 and 1.5 > 0.25, x > y.

Multiple choice

Directions: In the question given below, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 4x2 – 13x – 35 = 0 (ii) 2y2 – 39y + 154 = 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

Solving (i): 4x^2 - 13x - 35 = 0 -> (4x + 7)(x - 5) = 0, so x = 5 or -1.75. Solving (ii): 2y^2 - 39y + 154 = 0 -> (2y - 11)(y - 14) = 0, so y = 5.5 or 7. Comparing values: 5 < 5.5, 5 < 7, -1.75 < 5.5, -1.75 < 7. Thus, x < y.

Multiple choice

Directions: In the question given below, two equations (i) and (ii) are given, you have to solve both the equations and give answer accordingly. (i) 15x2 – 14x + 3 = 0 (ii) 8y2 – 26y + 21 = 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 15x^2 - 14x + 3 = 0, roots are 3/5 and 1/3 (0.6, 0.33). For 8y^2 - 26y + 21 = 0, roots are 7/4 and 3/2 (1.75, 1.5). Since all y values are greater than all x values, x < y.

Multiple choice

Directions: In the following question, equations numbered I and II are given. You have to solve both the equations and mark the answer as per the instructions. I. x2 + 25x + 156 = 0 II. y2 – 32y + 207 = 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
A Correct answer
Explanation

I: x^2 + 25x + 156 = 0 -> (x + 12)(x + 13) = 0, so x = -12, -13. II: y^2 - 32y + 207 = 0 -> (y - 9)(y - 23) = 0, so y = 9, 23. Comparing values, -12 < 9, -12 < 23, -13 < 9, -13 < 23. Thus, x < y.

Multiple choice

Directions: In the following question, equations numbered I and II are given. You have to solve both the equations and mark the answer as per the instructions. I. x2 – 48x + 567 = 0 II. y2 + 38y + 357 = 0

  1. x < y

  2. x = y or the relation cannot be established

  3. x ≥ y

  4. x > y

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

Equation I: x^2 - 48x + 567 = 0. Roots are 21 and 27. Equation II: y^2 + 38y + 357 = 0. Roots are -17 and -21. Since all values of x (21, 27) are greater than all values of y (-17, -21), x > y.

Multiple choice

Directions: In this question, two equations are given. You have to solve both the equations and mark the correct option. (i) 221x2 – 30x + 1 = 0 (ii) 483y2 – 44y + 1 = 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
A Correct answer
Explanation

Solving 221x^2 - 30x + 1 = 0: roots are 1/13 and 1/17. Solving 483y^2 - 44y + 1 = 0: roots are 1/21 and 1/23. Since 1/13 > 1/21 and 1/17 > 1/23, all x > y.

Multiple choice

Directions: In this question, two equations are given. You have to solve both the equations and mark the correct option. (i) x2 – 77x + 976 = 0 (ii) y2 + 76y + 915 = 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
A Correct answer
Explanation

For x^2 - 77x + 976 = 0, roots are (x-61)(x-16)=0, so x=61, 16. For y^2 + 76y + 915 = 0, roots are (y+61)(y+15)=0, so y=-61, -15. Since 61 > -15 and 16 > -61, x > y.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and give answer. I. x2 - 9x + 10 = -8 II. y2 - 18y + 60 = -20

  1. if x > y

  2. if x ≥ y

  3. if x < y

  4. if x ≤ y

  5. if x = y or Cannot be determined
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Equation I: x^2 - 9x + 18 = 0, so (x-3)(x-6)=0, x=3 or 6. Equation II: y^2 - 18y + 80 = 0, so (y-8)(y-10)=0, y=8 or 10. Comparing all pairs, x is always less than y.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and give answer. I. x2 - 16x + 60 = -3 II. y2 - 12y + 25 = -10

  1. if x > y

  2. if x ≥ y

  3. if x < y

  4. if x ≤ y

  5. if x = y or Cannot be determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

I: x^2 - 16x + 63 = 0. (x-7)(x-9) = 0, so x = 7, 9. II: y^2 - 12y + 35 = 0. (y-5)(y-7) = 0, so y = 5, 7. Comparing: x=7, y=5 (x>y); x=7, y=7 (x=y); x=9, y=5 (x>y); x=9, y=7 (x>y). Thus x >= y.

Multiple choice

Directions: In the following question, two equations, labeled (I) and (II) are given. You have to solve both the equations and give answer. (I) 5x2 = 15 + 74x (II) 96y + 19 = -5y2

  1. x > y

  2. x < y

  3. x ≥ y

  4. x ≤ y

  5. x = y or there is no relation between 'x' and 'y'
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Eq I: 5x^2 - 74x - 15 = 0. Roots: x = (74 +/- sqrt(5476 + 300))/10 = (74 +/- 76)/10. x = 15 or -0.2. Eq II: 5y^2 + 96y + 19 = 0. Roots: y = (-96 +/- sqrt(9216 - 380))/10 = (-96 +/- 94)/10. y = -0.2 or -19. Comparing roots: x values are 15, -0.2; y values are -0.2, -19. Thus x >= y.