Quantitative Aptitude

Quadratic Equations

367 Questions
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Quadratic Equations Questions

Multiple choice

Directions: In the given question, two equations numbered I and II are given. You have to solve both the equations and find the relationship between x and y. I. x2 + 7x + 12 = 0 II. y2 + 6y + 8 = 0

  1. x < y

  2. x = y or the relationship cannot be established

  3. x > y

  4. x ≥ y

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

I: x^2 + 7x + 12 = 0 => (x+3)(x+4) = 0 => x = -3, -4. II: y^2 + 6y + 8 = 0 => (y+2)(y+4) = 0 => y = -2, -4. Comparing values: -3 > -4, -3 < -2, -4 = -4. Since x can be greater than, less than, or equal to y, the relationship cannot 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. 10x2 - 7x + 1 = 0 II. 35y2 - 12y + 1 = 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

I: 10x^2 - 7x + 1 = 0 => (2x-1)(5x-1) = 0 => x = 0.5, 0.2. II: 35y^2 - 12y + 1 = 0 => (5y-1)(7y-1) = 0 => y = 0.2, 0.1428. Comparing roots: 0.5 > 0.2, 0.5 > 0.1428, 0.2 = 0.2, 0.2 > 0.1428. 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 select the correct answer. I. 2x2 + 18x + 36 = 0 II. 2y2 - 9y + 9 = 0

  1. x > y

  2. x ≥ y

  3. x < y

  4. x ≤ y

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

Equation I: 2x^2 + 18x + 36 = 0 -> x^2 + 9x + 18 = 0 -> (x+6)(x+3) = 0, so x = -6, -3. Equation II: 2y^2 - 9y + 9 = 0 -> (2y-3)(y-3) = 0, so y = 1.5, 3. Comparing the sets, all values of x are less than all values of y.

Multiple choice

Directions: Equation I and II are given. You have to solve both the equations and find out the relation, if any between x and y. I. 3x2 + 18x + 15 = 0 II. 14y2 + 3y = 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

I: 3x^2 + 18x + 15 = 0 -> x^2 + 6x + 5 = 0 -> (x+5)(x+1)=0, so x = -5, -1. II: 14y^2 + 3y - 2 = 0 -> 14y^2 + 7y - 4y - 2 = 0 -> 7y(2y+1) - 2(2y+1) = 0 -> (7y-2)(2y+1)=0, so y = 2/7, -1/2. Comparing values, x < y.

Multiple choice

Directions: Equation I and II are given. You have to solve both the equations and find out the relation, if any between x and y. I. 2x2 + 73x + 71 = 0 II. 11y + 6 = 21y2

  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

I: 2x^2 + 73x + 71 = 0. (2x + 71)(x + 1) = 0. Roots are x = -1, x = -35.5. II: 21y^2 - 11y - 6 = 0. (7y + 2)(3y - 3) is wrong. Using quadratic formula: y = (11 +/- sqrt(121 - 4*21*(-6))) / 42 = (11 +/- sqrt(121 + 504)) / 42 = (11 +/- 25) / 42. y = 36/42 = 6/7 or y = -14/42 = -1/3. Comparing roots: x = -1, -35.5 and y = 0.85, -0.33. In all cases, x < y.

Multiple choice

Directions: In the following question, you have to compare Quantity I and Quantity II and then determine which of the following options describes the comparison. a + b + c = 5 ab + bc + ca = 8 Quantity I: Value of a2 + b2 + c2 Quantity II: 6

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I Quantity II

  4. Quantity I ≥ Quantity II

  5. No relation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca), the value is 25 - 16 = 9. Since 9 > 6, Quantity I is greater than Quantity II.

Multiple choice

Directions: In the given question, equations numbered I and II are given. Solve both equations and give answer. I. 225y2 - 105y - 44 = 0 II. 210x2 - 319x + 121 = 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
C Correct answer
Explanation

Solving 225y^2 - 105y - 44 = 0 gives y = 0.733 or -0.266. Solving 210x^2 - 319x + 121 = 0 gives x = 1.1 or 0.523. Comparing the roots, x >= y is the correct relationship.

Multiple choice

Directions: In the given question, equations numbered I and II are given. Solve both equations and give answer. I. 12(x + 2)2 + 24(x + 2) + 9 = 0 II. 4y2 + 4y - 15 = 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
B Correct answer
Explanation

I: 12(x+2)^2 + 24(x+2) + 9 = 0. Let u = x+2. 12u^2 + 24u + 9 = 0. Divide by 3: 4u^2 + 8u + 3 = 0. (2u+1)(2u+3) = 0. u = -0.5, -1.5. x = -2.5, -3.5. II: 4y^2 + 4y - 15 = 0. (2y-3)(2y+5) = 0. y = 1.5, -2.5. Comparing: x <= y.

Multiple choice

Directions: In the given question, equations numbered I and II are given. Solve both equations and give answer. I. (x2 - 5x + 3) + (6x + 3 - 2x2) = 0 II. (2y2 - 11y + 5) - (y2 - 19y - 7) = 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
C Correct answer
Explanation

I: x^2 - 5x + 3 + 6x + 3 - 2x^2 = 0 => -x^2 + x + 6 = 0 => x^2 - x - 6 = 0 => (x-3)(x+2) = 0, so x = 3, -2. II: 2y^2 - 11y + 5 - y^2 + 19y + 7 = 0 => y^2 + 8y + 12 = 0 => (y+6)(y+2) = 0, so y = -6, -2. Comparing x={3, -2} and y={-6, -2}, 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 according to these codes: (a) If x > y (b) If x ≥ y (c) If x < y (d) If x ≤ y (e) If x = y or no relation can be established between x and y I. x(x + 4) = 5 II. y(y - 10) = -16

  1. (a)

  2. (b)

  3. (c)

  4. (d)

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

I: x^2 + 4x - 5 = 0 => (x+5)(x-1) = 0 => x = 1, -5. II: y^2 - 10y + 16 = 0 => (y-8)(y-2) = 0 => y = 8, 2. Comparing values: 1 < 8, 1 < 2, -5 < 8, -5 < 2. Thus x < y.

Multiple choice

Directions: Choose your answer as: One root of x2 - 9x + k = 0 is twice the other. Quantity A Quantity B K 27

  1. Quantity A is greater than quantity B.

  2. Quantity B is greater than quantity A.

  3. Both quantities are equal.

  4. The relationship between the quantities cannot be determined.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For x^2 - 9x + k = 0, let the roots be r and 2r. Sum of roots: 3r = 9, so r = 3. Product of roots: (r)(2r) = 2r^2 = k. Thus, k = 2(3^2) = 18. Since 18 < 27, Quantity B is greater.

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 - 6x = 2x - 4 Quantity A: 3x + 2 Quantity B: 4x + 6

  1. Quantity A is less than Quantity B.

  2. Quantity A is greater than Quantity B.

  3. The relationship cannot be determined.

  4. Quantity B is equal to Quantity A.

  5. Quantity B is greater than or equal to Quantity A.
Reveal answer Fill a bubble to check yourself
A Correct answer