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 mark the correct option accordingly. (i) x2 + 2x - 63 = 0 (ii) y2 + 27y + 182 = 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
A Correct answer
Explanation

Solving (i) x^2 + 2x - 63 = 0 gives (x+9)(x-7)=0, so x = 7 or -9. Solving (ii) y^2 + 27y + 182 = 0 gives (y+13)(y+14)=0, so y = -13 or -14. Since both values of x are greater than both values of y, x > y.

Multiple choice

In this question, two equations numbered I and II are given. You have to solve both the equations and mark the correct answer. I. x2 - 49 = 0 II. y2 - 9y + 20 = 0

  1. if x > y

  2. if x < y

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

  4. if x ≤ y

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

I: x^2 = 49 implies x = 7 or -7. II: y^2 - 9y + 20 = 0 factors to (y-4)(y-5) = 0, so y = 4 or 5. Comparing x and y values, sometimes x > y (7 > 4) and sometimes x < y (-7 < 4), so no relationship can be established.

Multiple choice

Directions: In the following question, two equations numbered I and II are given. You have to solve both the equations and then choose the correct option. I. x2 + 11x + 28 = 0 II. 2y2 + 13y + 20 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

I: x^2 + 11x + 28 = 0 -> (x+4)(x+7) = 0 -> x = -4, -7. II: 2y^2 + 13y + 20 = 0 -> (2y+5)(y+4) = 0 -> y = -2.5, -4. Comparing: -4 <= -2.5 and -4 <= -4, -7 <= -2.5 and -7 <= -4. Thus 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 then choose the correct option. I. 6x2 + 29x + 35 = 0 II. 3y2 + 19y + 30 = 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 I: 6x^2 + 29x + 35 = 0 -> (2x+5)(3x+7) = 0, so x = -2.5 or -2.33. For II: 3y^2 + 19y + 30 = 0 -> (3y+10)(y+3) = 0, so y = -3.33 or -3. Comparing values, -2.5 > -3.33 and -2.5 > -3, and -2.33 > -3.33 and -2.33 > -3. Thus x > y.

Multiple choice

Directions: Equations I and II are given. You have to solve both the equations and choose the correct answer. I. 10x2 = 4 - 6x II. 3y2 + 7y + 4 = 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

Eq I: 10x^2 + 6x - 4 = 0 -> 5x^2 + 3x - 2 = 0 -> (5x-2)(x+1) = 0. x = 0.4 or -1. Eq II: 3y^2 + 7y + 4 = 0 -> (3y+4)(y+1) = 0. y = -4/3 or -1. Comparing values, x >= y.

Multiple choice

Directions: Equations I and II are given. You have to solve both the equations and choose the correct answer. I. 50x2 = 51x - 1 II. 2y2 + 31y + 29 = 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
A Correct answer
Explanation

I: 50x^2 - 51x + 1 = 0. Roots are (51 +/- sqrt(51^2 - 200))/100 = (51 +/- 49)/100. x = 1 or 0.02. II: 2y^2 + 31y + 29 = 0. Roots are (-31 +/- sqrt(31^2 - 232))/4 = (-31 +/- 27)/4. y = -1 or -14.5. Comparing roots, x > y.

Multiple choice

Directions: In the question, two equations numbered I and II are given. You have to find the relationship between them by solving both the equations and then mark the answer accordingly. I. 18x2 + 21x + 3 = 0 II. 45y2 + 24y + 3 = 0

  1. x > y

  2. x < y

  3. x = y or no relation can be established

  4. x ≥ y

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

Equation I: 18x^2 + 21x + 3 = 0 -> 6x^2 + 7x + 1 = 0 -> (6x+1)(x+1) = 0. Roots are -1/6, -1. Equation II: 45y^2 + 24y + 3 = 0 -> 15y^2 + 8y + 1 = 0 -> (5y+1)(3y+1) = 0. Roots are -1/5, -1/3. Comparing the sets {-1, -0.166} and {-0.333, -0.2}, no consistent relationship exists.

Multiple choice

Directions: In the question, two equations numbered I and II are given. You have to find the relationship between them by solving both the equations and then mark the answer accordingly. I. 8x2 + 20x + 12 = 0 II. 4y2 + 39y + 56 = 0

  1. x > y

  2. x < y

  3. x = y or no relation can be established

  4. x ≥ y

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

Equation I: 8x^2 + 20x + 12 = 0 -> 2x^2 + 5x + 3 = 0 -> (2x+3)(x+1) = 0. Roots are x = -1.5, -1. Equation II: 4y^2 + 39y + 56 = 0 -> (4y+7)(y+8) = 0. Roots are y = -1.75, -8. Comparing roots, -1 > -1.75 and -1 > -8, and -1.5 > -1.75 and -1.5 > -8. Thus, x > y.

Multiple choice

Directions: In the question, two equations numbered I and II are given. You have to find the relationship between them by solving both the equations and then mark the answer accordingly. I. 12x2 - 16x + 4 = 0 II. 60y2 - 60y + 15 = 0

  1. x > y

  2. x < y

  3. x = y or no relation can be established

  4. x ≥ y

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

Equation I: 12x^2 - 16x + 4 = 0 simplifies to 3x^2 - 4x + 1 = 0, which factors to (3x-1)(x-1)=0, giving x = 1/3 or 1. Equation II: 60y^2 - 60y + 15 = 0 simplifies to 4y^2 - 4y + 1 = 0, which is (2y-1)^2 = 0, giving y = 1/2. Comparing the roots, x=1 > y=1/2 and x=1/3 < y=1/2, so no consistent relationship exists.

Multiple choice

Directions: In the problem, two equations numbered I and II are given. You have to find the relationship between them by solving both the equations and then mark the answer accordingly. I. x2 + 8x + 15 = 0 II. y2 - 14y + 45 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

I: x^2 + 8x + 15 = 0 -> (x+3)(x+5) = 0 -> x = -3, -5. II: y^2 - 14y + 45 = 0 -> (y-5)(y-9) = 0 -> y = 5, 9. Clearly, all values of x are less than all values of 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 according to these codes: (a) If x > y (b) If x ≥ y (c) If x 2 - 28x + 192 = 0 II. y2 + 14y - 95 = 0

  1. (a)

  2. (b)

  3. (c)

  4. (d)

  5. (e)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The first equation factors as (x - 12)(x - 16) = 0, so x is 12 or 16. The second equation factors as (y - 5)(y + 19) = 0, so y is 5 or -19. Every possible x is greater than every possible y, hence x > y.