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 given question, two equations are given. You have to solve both the equations and mark the correct option. (i) x2 - 44.8x + 500.55 = 0 (ii) y2 - 28.8y + 206.15 = 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 - 44.8x + 500.55 = 0, roots are approx 15.7 and 29.1. For y^2 - 28.8y + 206.15 = 0, roots are approx 10.7 and 18.1. Comparing roots, x > y holds.

Multiple choice

Directions: In the given question, two equations are given. You have to solve both the equations and mark the correct option. (i) X2 - 39X + 380 = 0 (ii) Y2 - 41Y + 420 = 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
D Correct answer
Explanation

Solving X^2 - 39X + 380 = 0: (X-19)(X-20) = 0, so X = 19, 20. Solving Y^2 - 41Y + 420 = 0: (Y-20)(Y-21) = 0, so Y = 20, 21. Comparing values, 19 < 20, 19 < 21, 20 = 20, 20 < 21. Thus 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) 6x2 - 43x + 77 = 0 (ii) 10y2 - 327y + 2501 = 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
B Correct answer
Explanation

6x^2 - 43x + 77 = 0. Roots are (43 +/- sqrt(1849 - 1848))/12 = (43 +/- 1)/12. x = 44/12 = 3.66 or x = 42/12 = 3.5. 10y^2 - 327y + 2501 = 0. Roots are (327 +/- sqrt(106929 - 100040))/20 = (327 +/- sqrt(6889))/20 = (327 +/- 83)/20. y = 410/20 = 20.5 or y = 244/20 = 12.2. Clearly 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) 77x2 - 1078x + 3465 = 0 (ii) y2 - 39y + 338 = 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
B Correct answer
Explanation

Eq (i): 77x^2 - 1078x + 3465 = 0. Divide by 77: x^2 - 14x + 45 = 0. (x-9)(x-5)=0, so x=9 or 5. Eq (ii): y^2 - 39y + 338 = 0. (y-13)(y-26)=0, so y=13 or 26. Comparing roots: 9 < 13, 9 < 26, 5 < 13, 5 < 26. Thus x < y.

Multiple choice

Quantity I. 'x': x2 + x - 6 = 0 Quantity II. 'y': y2 + 7y + 12 = 0

  1. a) Quantity I > Quantity II

  2. b) Quantity I < Quantity II

  3. c) Quantity I ≥ Quantity II

  4. d) Quantity I ≤ Quantity II

  5. e) Quantity I = Quantity II or No relation
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Quantity I: x^2 + x - 6 = 0 => (x+3)(x-2) = 0 => x = -3, 2. Quantity II: y^2 + 7y + 12 = 0 => (y+3)(y+4) = 0 => y = -3, -4. Comparing sets {-3, 2} and {-3, -4}: 2 > -3, 2 > -4, -3 = -3, -3 > -4. In all cases, x >= y.

Multiple choice

One of the two students, while solving a quadratic equation in x, copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of x2 correctly and got his roots as -6 and 1 respectively. The correct roots are

  1. 3, -2

  2. -3, 2
    • 6, -1

  4. 6, -1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let the equation be x^2 - Sx + P = 0. The first student got roots 3 and 2, so the sum S = 5 is correct, but the product P is wrong. The second student got roots -6 and 1, so the product P = -6 is correct, but the sum S = -5 is wrong. The correct equation is x^2 - 5x - 6 = 0. Factoring gives (x-6)(x+1) = 0, so roots are 6 and -1.

Multiple choice

Solve the following inequalities: 3| x – 1| + x2 – 7 > 0

  1. x > -1

  2. x < - 1

  3. x > 2

  4. Both (b) and (c)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Solve 3|x-1| + x^2 - 7 > 0. Case 1: x >= 1, 3(x-1) + x^2 - 7 > 0 -> x^2 + 3x - 10 > 0 -> (x+5)(x-2) > 0. For x >= 1, this holds for x > 2. Case 2: x < 1, 3(1-x) + x^2 - 7 > 0 -> x^2 - 3x - 4 > 0 -> (x-4)(x+1) > 0. For x < 1, this holds for x < -1. Combining gives x < -1 or x > 2.

Multiple choice

In each question two equations numbered I and II are given. You have to solve both the equations and mark the answer. I. x2 – 12x + 32 = 0 II. 2y2 – 9y + 10 = 0

  1. A. if x > y

  2. B. if x ≥ y

  3. C. if x < y

  4. D. if x ≤ y

  5. E. if x = y or the relation between x and y can't be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

I. x^2 - 12x + 32 = 0 => (x-8)(x-4) = 0, so x = 8, 4. II. 2y^2 - 9y + 10 = 0 => (2y-5)(y-2) = 0, so y = 2.5, 2. Comparing values, x > y in all cases.

Multiple choice

In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. I. x2 – 0.5x – 39 = 0 II. y2 – 15.5y + 60 = 0

  1. if x > y

  2. if x ≤ y

  3. if x ≥ y

  4. if x < y

  5. if x = y or relationship between x and y can't be established
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

I: x^2 - 0.5x - 39 = 0. Roots using quadratic formula: (0.5 +/- sqrt(0.25 + 156)) / 2 = (0.5 +/- 12.5) / 2. x = 6.5 or -6. II: y^2 - 15.5y + 60 = 0. Roots: (15.5 +/- sqrt(240.25 - 240)) / 2 = (15.5 +/- 0.5) / 2. y = 8 or 7.5. Comparing x and y, x < y.