Quantitative Aptitude

Quadratic Equations

367 Questions
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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. (I) x2 - 40x + 39 = 0 (II) y2 + y - 2 = 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

I: x^2 - 40x + 39 = 0 -> (x-39)(x-1) = 0 -> x = 39, 1. II: y^2 + y - 2 = 0 -> (y+2)(y-1) = 0 -> y = -2, 1. Comparing roots: x=39, y=-2 (x>y); x=39, y=1 (x>y); x=1, y=-2 (x>y); x=1, y=1 (x=y). Thus x >= y.

Multiple choice

One of the roots of x2 - 20x + k = 0, is 9. Quantity A k Quantity B 100

  1. Quantity A is greater.

  2. Quantity 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
B Correct answer
Explanation

If 9 is a root, 9^2 - 20(9) + k = 0. 81 - 180 + k = 0, so k = 99. Quantity A is 99, Quantity B is 100. Quantity B is greater.

Multiple choice

Following question contains two equations as I and II. You have to solve both equations and determine the relationship between them. I) 2x2 + 12x + 18 = 0 II) 3y2 + 13y + 12 = 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
D Correct answer
Explanation

I) 2x^2 + 12x + 18 = 0 -> 2(x^2 + 6x + 9) = 0 -> 2(x+3)^2 = 0 -> x = -3. II) 3y^2 + 13y + 12 = 0 -> (3y+4)(y+3) = 0 -> y = -4/3 or y = -3. Since x = -3 and y is -1.33 or -3, x <= y.

Multiple choice

x2(x - 3) = 0 Quantity A x Quantity B 2

  1. Quantity A is greater.

  2. Quantity 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

x^2(x-3) = 0 implies x=0 or x=3. If x=0, Quantity A < Quantity B. If x=3, Quantity A > Quantity B. Since the value of x is not unique, the relationship cannot be determined.

Multiple choice

In the following questions two equations numbered I and II are given. You have to solve both the equations and choose the correct option. I. x2-15x +36 = 0 II. 10y2 +31y -63 = 0

  1. x < y

  2. x > y

  3. x ≤ y

  4. x ≥ y

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

Solving I: x^2 - 15x + 36 = 0 gives (x-3)(x-12) = 0, so x = 3, 12. Solving II: 10y^2 + 31y - 63 = 0, using the quadratic formula or factoring, gives (2y + 9)(5y - 7) = 0, so y = -4.5, 1.4. Comparing the sets, all values of x are greater than all values of y.

Multiple choice

In the following questions two equations numbered I and II are given. You have to solve both the equations and choose the correct option. I. x2 +15x +54 = 0 II. y2-5y -66 = 0

  1. x < y

  2. x > y

  3. x ≤ y

  4. x ≥ y

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

I: x^2 + 15x + 54 = 0 => (x+6)(x+9) = 0 => x = -6, -9. II: y^2 - 5y - 66 = 0 => (y-11)(y+6) = 0 => y = 11, -6. Comparing: -6 <= 11, -6 <= -6, -9 <= 11, -9 <= -6. Thus, x <= y.

Multiple choice

Following question contains two equations as I and II. You have to solve both equations and determine the relationship between them. I) x2– 32x + 247 = 0 II) y2– 35y + 304 = 0

  1. x > y

  2. x ≥ y

  3. x = y or relationship can't be determined.

  4. x < y

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

For I: x^2 - 32x + 247 = 0, the roots are 13 and 19. For II: y^2 - 35y + 304 = 0, the roots are 16 and 19. Comparing the sets {13, 19} and {16, 19}, we see that x can be less than, equal to, or greater than y depending on the specific root chosen, so the relationship cannot be determined.

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. Consider the following curves on the coordinate plane: Curve A: x = 3y2 - 2 and Curve B: x = -2y2 + 5. Quantity I: The x-coordinate of the point where Curve A has its minimum value Quantity II: The x-coordinate of the point where Curve B has its maximum value

  1. Quantity II > Quantity I

  2. Quantity I > Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity II ≥ Quantity I

  5. Quantity I = quantity II, or No relation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Curve A: x = 3y^2 - 2. The minimum x occurs at y = 0, so x = -2. Curve B: x = -2y^2 + 5. The maximum x occurs at y = 0, so x = 5. Quantity I = -2, Quantity II = 5. Thus, Quantity II > Quantity I.

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 - 26x + 165 = 0 II. y2 - 16y = -63

  1. if x > y

  2. if x ≥ y

  3. if x < y

  4. if x ≤ y

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

I: x^2 - 26x + 165 = 0 -> (x-11)(x-15) = 0, x = 11, 15. II: y^2 - 16y + 63 = 0 -> (y-7)(y-9) = 0, y = 7, 9. Since all x values (11, 15) are greater than all y values (7, 9), x > y.

Multiple choice

y is a positive integer. Quantity A y2 + 2 Quantity B 3y - 1

  1. Quantity A is greater.

  2. Quantity 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
A Correct answer
Explanation

Compare y^2 + 2 and 3y - 1. For y=1, A=3, B=2 (A>B). For y=2, A=6, B=5 (A>B). For y=3, A=11, B=8 (A>B). Since y^2 - 3y + 3 = (y - 1.5)^2 + 0.75, which is always positive, y^2 + 2 is always greater than 3y - 1.

Multiple choice

In the given question, two equations, numbered I and II, are given. Solve both the equations to get the appropriate answer. I. 2x2 - 87x + 43 = 0 II. y2 - 2y - 63 = 0

  1. If x > y

  2. If x ≥ y

  3. If x < y

  4. If x = y or the relationship cannot be established

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

Solving 2x^2 - 87x + 43 = 0 gives x = 43 or 0.5. Solving y^2 - 2y - 63 = 0 gives y = 9 or -7. Comparing the sets {43, 0.5} and {9, -7}, the relationship cannot be established.

Multiple choice

(x - 4)(x + 2) = (3x - 5) Quantity A x Quantity B 3

  1. Quantity A is greater.

  2. Quantity 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

(x-4)(x+2) = x^2 - 2x - 8 = 3x - 5. x^2 - 5x - 3 = 0. Using the quadratic formula, x = (5 +/- sqrt(25 - 4*1*-3)) / 2 = (5 +/- sqrt(37)) / 2. sqrt(37) is between 6 and 7. x1 approx (5+6.08)/2 = 5.54. x2 approx (5-6.08)/2 = -0.54. Since x can be either value, it cannot be determined if x > 3 or x < 3.

Multiple choice

x2 + 2y2 = 1 x = 1/2 Column A Column B |y| 1/2 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
A Correct answer
Explanation

Given x^2 + 2y^2 = 1 and x = 1/2, then (1/4) + 2y^2 = 1, so 2y^2 = 3/4, y^2 = 3/8. Thus |y| = sqrt(3/8). Since sqrt(3/8) is approximately sqrt(0.375) which is greater than 0.5 (1/2), Column A is greater.