Quantitative Aptitude

Quadratic Equations

367 Questions
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Quadratic Equations Questions

Multiple choice

(x - 1) (x + 2) (2x + 3) (3x - 6) (x + 1) = 0 Column A Column B The number of possible values of x that are integers. 4 Compare the two quantities to decide whether

  1. the quantity in Column A is greater

  2. the quantity in Column B is greater

  3. the two quantities are equal

  4. the relationship cannot be determined from the information given
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The roots of (x-1)(x+2)(2x+3)(3x-6)(x+1) = 0 are 1, -2, -1.5, 2, -1. The integers are 1, -2, 2, -1. There are 4 integer roots. Column A is 4, Column B is 4. They are equal.

Multiple choice

x2 - 6x - 567 = 0. It is given that x is a real number. Column A Column B x 27 Compare the two quantities to decide whether

  1. the quantity in Column A is greater

  2. the quantity in Column B is greater

  3. the two quantities are equal

  4. the relationship cannot be determined from the information given
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The quadratic equation x^2 - 6x - 567 = 0 can be solved using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a. This yields x = (6 +/- sqrt(36 + 2268)) / 2 = (6 +/- sqrt(2304)) / 2 = (6 +/- 48) / 2, resulting in x = 27 or x = -21. Since x can be either 27 or -21, the relationship between x and 27 cannot be determined.

Multiple choice

Directions: The following problem has a question and two statements which are labelled (A) and (B). Use the data given in (A) and (B) to decide whether the statements are sufficient to answer the question. Then mark answer: Is X positive? (A) X2 - 1 = 0 (B) X3 + 1 = 0

  1. If the question can be solved using any one of the statements.

  2. If the question can be solved using either of the statements.

  3. If the question can be solved using both but not either alone.

  4. If the question cannot be solved using the given statements.
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

Directions: In the following question, two equations are given. You have to solve both the equations and then choose the correct option. 5x2 − 310x + 3000 = 0 7y2 + 80y - 1500 = 0

  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
A Correct answer
Explanation

5x^2 - 310x + 3000 = 0 => x^2 - 62x + 600 = 0. (x-50)(x-12)=0. x=50, 12. 7y^2 + 80y - 1500 = 0. y = [-80 +/- sqrt(6400 - 4*7*-1500)] / 14 = [-80 +/- sqrt(6400 + 42000)] / 14 = [-80 +/- 220] / 14. y = 140/14 = 10 or -300/14 = -21.4. Comparing roots: 50 > 10, 50 > -21.4, 12 > 10, 12 > -21.4. Thus x > y.

Multiple choice

During a math competition, Garry and Daniel were given a polynomial problem to solve. If one zero of the polynomial f(x) = x2 - (k - 3)x + 12 is triple the other zero, what is the sum of all the possible values of k?

  1. -11

  2. 6

  3. 7

  4. 9

  5. 11
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let roots be a and 3a. Sum of roots = 4a = k - 3. Product of roots = 3a^2 = 12, so a^2 = 4, a = 2 or -2. If a = 2, 4(2) = k - 3 => k = 11. If a = -2, 4(-2) = k - 3 => k = -5. Sum of possible k values = 11 + (-5) = 6.

Multiple choice

Directions: In the following question, two equations are given. You have to solve both the equations and then choose the correct option. 2x2 + 225x + 6175 = 0 4y2 + 183y + 1620 = 0

  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
B Correct answer
Explanation

For 2x^2 + 225x + 6175 = 0, the roots are -65 and -47.5. For 4y^2 + 183y + 1620 = 0, the roots are -15 and -27. Since all x values are smaller than all y values, x < y.

Multiple choice

(x – 1)(x + 1) = 0 Quantity A Quantity B x 0

  1. Quantity A is greater than Quantity B

  2. Quantity A is smaller than Quantity B

  3. Quantity A is equal to Quantity B

  4. The relationship cannot be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

(x-1)(x+1) = 0 implies x = 1 or x = -1. Since x can be either 1 or -1, and Quantity B is 0, we cannot determine if A > B, A < B, or A = B.

Multiple choice

Directions: In the following item, Quantity I and Quantity II are given. Determine the relationship between the quantities and choose the appropriate option. 3x2 + 25x - 28 = 0 Quantity I: x - 5 Quantity II: x2 + 5

  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 No relation
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Solving 3x^2 + 25x - 28 = 0 gives (3x + 28)(x - 1) = 0, so x = 1 or x = -28/3. If x = 1, Q1 = -4, Q2 = 6 (Q1 < Q2). If x = -28/3, Q1 = -43/3, Q2 = 81.11 (Q1 < Q2). In both cases, Q1 < Q2.

Multiple choice

Directions: There are two equations given in the problem. Solve the equations and answer accordingly. I. 3x2 + 11x + 10 = 0 II. 2y2 + 13y + 21 = 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

Solving 3x^2 + 11x + 10 = 0 gives (3x + 5)(x + 2) = 0, so x = -5/3 or -2. Solving 2y^2 + 13y + 21 = 0 gives (2y + 7)(y + 3) = 0, so y = -3.5 or -3. Comparing the values, -5/3 (-1.67) and -2 are both greater than -3.5 and -3, so x > y.

Multiple choice

Directions: There are two equations given in the problem. Solve the equations and answer accordingly. I. 7x2 - 25x - 12 = 0 II. 3y2 + 10y + 8 = 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: 7x^2 - 25x - 12 = 0 => 7x^2 - 28x + 3x - 12 = 0 => 7x(x-4) + 3(x-4) = 0 => (7x+3)(x-4) = 0. x = 4, -3/7. II: 3y^2 + 10y + 8 = 0 => 3y^2 + 6y + 4y + 8 = 0 => 3y(y+2) + 4(y+2) = 0 => (3y+4)(y+2) = 0. y = -4/3, -2. Comparing: 4 > -4/3, 4 > -2, -3/7 > -4/3, -3/7 > -2. Thus x > y.

Multiple choice

Solve the equations given below and choose the answer from the given alternatives. 9x2 + 9 = 30x 12y2 – 9 = -33y

  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

9x^2 - 30x + 9 = 0 => 3x^2 - 10x + 3 = 0 => (3x-1)(x-3) = 0. x = 1/3 or 3. 12y^2 + 33y - 9 = 0 => 4y^2 + 11y - 3 = 0 => (4y-1)(y+3) = 0. y = 1/4 or -3. Comparing values, x can be 3 or 1/3, y can be 0.25 or -3. In all cases, x > y.