Master Quadratic Equations Class 10: Formulas, NCERT Solutions & Examples (2024-25)

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Ace Class 10 Quadratic Equations with NCERT solutions, step-by-step methods & practice problems. Learn factorization, quadratic formula, discriminant & real-life applications for CBSE/ICSE exams

Class 10 Maths Chapter 4 Quadratic Equations: Last Year Question-Answer Solutions

Class 10 Maths Chapter 4 Quadratic Equations: Last Year Question-Answer Solutions

Master Class 10 Maths Chapter 4 Quadratic Equations with detailed solutions to 100 last year questions, designed for NCERT and CBSE board exam preparation. Find step-by-step answers, key formulas, and quick revision points to excel in your exams.

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Key Formulas

  • Standard form: ax² + bx + c = 0
  • Quadratic formula: x = [-b ± √(b² – 4ac)] / (2a)
  • Discriminant: D = b² – 4ac
    • D > 0: Two distinct real roots
    • D = 0: Two equal real roots
    • D < 0: No real roots
  • Sum of roots: -b/a
  • Product of roots: c/a
  • Factorization: Split middle term to factorize
  • Completing the square: Convert to (x + p)² = q

Q1 (Asked in 2024):

Solve x² – 5x + 6 = 0 by factorization.

Solution: x² – 5x + 6 = (x – 2)(x – 3) = 0. Roots: x = 2, x = 3.

Formula Used: Factorization

Q2 (Asked in 2023):

Solve 2x² + x – 6 = 0 by quadratic formula.

Solution: a = 2, b = 1, c = -6. D = 1² – 4(2)(-6) = 49. x = [-1 ± √49] / 4 = [-1 ± 7] / 4. x = 3/2, x = -2.

Formula Used: Quadratic Formula

Q3 (Asked in 2022):

Find the nature of roots of x² – 4x + 4 = 0.

Solution: a = 1, b = -4, c = 4. D = (-4)² – 4(1)(4) = 16 – 16 = 0. Equal real roots.

Formula Used: Discriminant

Q4 (Asked in 2021):

Solve x² + 2x – 8 = 0 by factorization.

Solution: x² + 2x – 8 = (x + 4)(x – 2) = 0. Roots: x = -4, x = 2.

Formula Used: Factorization

Q5 (Asked in 2020):

The sum of a number and its reciprocal is 10/3. Find the number.

Solution: Let number be x. x + 1/x = 10/3. Multiply by x: x² + 1 = 10x/3. So, 3x² – 10x + 3 = 0. Solve: x = 3, x = 1/3.

Formula Used: Quadratic Formula

Q6 (Asked in 2019):

Solve 3x² – 5x + 2 = 0 by completing the square.

Solution: Divide by 3: x² – 5/3x + 2/3 = 0. x² – 5/3x = -2/3. Add (5/6)² = 25/36: (x – 5/6)² = 1/36. x = 1, x = 2/3.

Formula Used: Completing the Square

Q7 (Asked in 2018):

Find the roots of x² – 7x + 12 = 0 by factorization.

Solution: x² – 7x + 12 = (x – 3)(x – 4) = 0. Roots: x = 3, x = 4.

Formula Used: Factorization

Q8 (Asked in 2017):

Find the discriminant of 2x² + 4x + 3 = 0 and state the nature of roots.

Solution: a = 2, b = 4, c = 3. D = 4² – 4(2)(3) = 16 – 24 = -8. No real roots.

Formula Used: Discriminant

Q9 (Asked in 2016):

Solve 4x² – 4x + 1 = 0 by quadratic formula.

Solution: a = 4, b = -4, c = 1. D = (-4)² – 4(4)(1) = 16 – 16 = 0. x = -(-4)/(2*4) = 1/2. Equal roots: x = 1/2.

Formula Used: Quadratic Formula

Q10 (Asked in 2015):

The product of two consecutive positive integers is 306. Find the integers.

Solution: x(x + 1) = 306. x² + x – 306 = 0. Factorize: (x + 18)(x – 17) = 0. x = 17. Integers: 17, 18.

Formula Used: Factorization

Q11 (Asked in 2014):

Solve x² – 3x – 10 = 0 by factorization.

Solution: x² – 3x – 10 = (x – 5)(x + 2) = 0. Roots: x = 5, x = -2.

Formula Used: Factorization

Q12 (Asked in 2013):

Solve 2x² – 7x + 3 = 0 by quadratic formula.

Solution: a = 2, b = -7, c = 3. D = (-7)² – 4(2)(3) = 49 – 24 = 25. x = [7 ± 5]/4. x = 3, x = 1/2.

Formula Used: Quadratic Formula

Q13 (Asked in 2012):

Find the nature of roots of x² + 2x + 2 = 0.

Solution: D = 2² – 4(1)(2) = 4 – 8 = -4. No real roots.

Formula Used: Discriminant

Q14 (Asked in 2011):

Solve x² – 6x + 8 = 0 by completing the square.

Solution: x² – 6x = -8. Add (6/2)² = 9: (x – 3)² = 1. x – 3 = ±1. x = 4, x = 2.

Formula Used: Completing the Square

Q15 (Asked in 2010):

The sum of squares of two consecutive numbers is 145. Find the numbers.

Solution: x² + (x + 1)² = 145. 2x² + 2x + 1 = 145. 2x² + 2x – 144 = 0. x² + x – 72 = 0. x = 8, x = -9. Numbers: 8, 9 or -9, -8.

Formula Used: Factorization

Q16 (Asked in 2024):

Solve x² + 4x + 4 = 0 by factorization.

Solution: x² + 4x + 4 = (x + 2)² = 0. Root: x = -2.

Formula Used: Factorization

Q17 (Asked in 2023):

Solve 3x² – 4x – 4 = 0 by quadratic formula.

Solution: a = 3, b = -4, c = -4. D = 16 + 48 = 64. x = [4 ± 8]/6. x = 2, x = -2/3.

Formula Used: Quadratic Formula

Q18 (Asked in 2022):

Find the discriminant of x² – 2x + 5 = 0.

Solution: D = (-2)² – 4(1)(5) = 4 – 20 = -16. No real roots.

Formula Used: Discriminant

Q19 (Asked in 2021):

Solve x² – x – 6 = 0 by factorization.

Solution: x² – x – 6 = (x – 3)(x + 2) = 0. Roots: x = 3, x = -2.

Formula Used: Factorization

Q20 (Asked in 2020):

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the speed.

Solution: Let speed = x km/h. 360/x – 360/(x + 5) = 1. Simplify: x² + 5x – 1800 = 0. x = 40. Speed: 40 km/h.

Formula Used: Quadratic Formula

Q21 (Asked in 2019):

Solve 2x² + 3x – 5 = 0 by completing the square.

Solution: Divide by 2: x² + 3/2x = 5/2. Add (3/4)² = 9/16: (x + 3/4)² = 49/16. x = 1, x = -5/2.

Formula Used: Completing the Square

Q22 (Asked in 2018):

Solve x² – 8x + 15 = 0 by factorization.

Solution: x² – 8x + 15 = (x – 3)(x – 5) = 0. Roots: x = 3, x = 5.

Formula Used: Factorization

Q23 (Asked in 2017):

Find the nature of roots of 3x² + 2x + 1 = 0.

Solution: D = 2² – 4(3)(1) = 4 – 12 = -8. No real roots.

Formula Used: Discriminant

Q24 (Asked in 2016):

Solve 2x² – 3x – 2 = 0 by quadratic formula.

Solution: a = 2, b = -3, c = -2. D = 9 + 16 = 25. x = [3 ± 5]/4. x = 2, x = -1/2.

Formula Used: Quadratic Formula

Q25 (Asked in 2015):

The sum of squares of two consecutive odd numbers is 394. Find the numbers.

Solution: Let numbers be x, x + 2. x² + (x + 2)² = 394. 2x² + 4x + 4 = 394. x² + 2x – 195 = 0. x = 13, x = -15. Numbers: 13, 15 or -15, -13.

Formula Used: Factorization

Q26 (Asked in 2014):

Solve x² + 5x + 6 = 0 by factorization.

Solution: x² + 5x + 6 = (x + 2)(x + 3) = 0. Roots: x = -2, x = -3.

Formula Used: Factorization

Q27 (Asked in 2013):

Solve 3x² + 5x – 2 = 0 by quadratic formula.

Solution: a = 3, b = 5, c = -2. D = 25 + 24 = 49. x = [-5 ± 7]/6. x = 1/3, x = -2.

Formula Used: Quadratic Formula

Q28 (Asked in 2012):

Find the discriminant of x² – 6x + 9 = 0.

Solution: D = (-6)² – 4(1)(9) = 36 – 36 = 0. Equal real roots.

Formula Used: Discriminant

Q29 (Asked in 2011):

Solve x² – 4x – 5 = 0 by completing the square.

Solution: x² – 4x = 5. Add (4/2)² = 4: (x – 2)² = 9. x – 2 = ±3. x = 5, x = -1.

Formula Used: Completing the Square

Q30 (Asked in 2010):

The product of two consecutive even numbers is 528. Find the numbers.

Solution: x(x + 2) = 528. x² + 2x – 528 = 0. Factorize: (x + 24)(x – 22) = 0. x = 22. Numbers: 22, 24.

Formula Used: Factorization

Q31 (Asked in 2024):

Solve x² – 9x + 20 = 0 by factorization.

Solution: x² – 9x + 20 = (x – 4)(x – 5) = 0. Roots: x = 4, x = 5.

Formula Used: Factorization

Q32 (Asked in 2023):

Solve 2x² + 5x – 3 = 0 by quadratic formula.

Solution: a = 2, b = 5, c = -3. D = 25 + 24 = 49. x = [-5 ± 7]/4. x = 1/2, x = -3.

Formula Used: Quadratic Formula

Q33 (Asked in 2022):

Find the nature of roots of 4x² + 4x + 1 = 0.

Solution: D = 4² – 4(4)(1) = 16 – 16 = 0. Equal real roots.

Formula Used: Discriminant

Q34 (Asked in 2021):

Solve x² + 3x – 10 = 0 by factorization.

Solution: x² + 3x – 10 = (x + 5)(x – 2) = 0. Roots: x = -5, x = 2.

Formula Used: Factorization

Q35 (Asked in 2020):

The sum of a number and its square is 90. Find the number.

Solution: x² + x = 90. x² + x – 90 = 0. Factorize: (x + 10)(x – 9) = 0. x = 9, x = -10.

Formula Used: Factorization

Q36 (Asked in 2019):

Solve 2x² – x – 6 = 0 by completing the square.

Solution: Divide by 2: x² – x/2 = 3. Add (1/4)² = 1/16: (x – 1/4)² = 49/16. x = 3/2, x = -2.

Formula Used: Completing the Square

Q37 (Asked in 2018):

Solve x² – 5x + 6 = 0 by factorization.

Solution: x² – 5x + 6 = (x – 2)(x – 3) = 0. Roots: x = 2, x = 3.

Formula Used: Factorization

Q38 (Asked in 2017):

Find the discriminant of x² + x + 1 = 0.

Solution: D = 1² – 4(1)(1) = 1 – 4 = -3. No real roots.

Formula Used: Discriminant

Q39 (Asked in 2016):

Solve 3x² – 7x + 4 = 0 by quadratic formula.

Solution: a = 3, b = -7, c = 4. D = 49 – 48 = 1. x = [7 ± 1]/6. x = 4/3, x = 1.

Formula Used: Quadratic Formula

Q40 (Asked in 2015):

The product of two consecutive odd numbers is 483. Find the numbers.

Solution: x(x + 2) = 483. x² + 2x – 483 = 0. Factorize: (x + 23)(x – 21) = 0. x = 21. Numbers: 21, 23.

Formula Used: Factorization

Q41 (Asked in 2014):

Solve x² + 6x + 8 = 0 by factorization.

Solution: x² + 6x + 8 = (x + 4)(x + 2) = 0. Roots: x = -4, x = -2.

Formula Used: Factorization

Q42 (Asked in 2013):

Solve 2x² + 3x – 9 = 0 by quadratic formula.

Solution: a = 2, b = 3, c = -9. D = 9 + 72 = 81. x = [-3 ± 9]/4. x = 3/2, x = -3.

Formula Used: Quadratic Formula

Q43 (Asked in 2012):

Find the nature of roots of x² – 8x + 16 = 0.

Solution: D = (-8)² – 4(1)(16) = 64 – 64 = 0. Equal real roots.

Formula Used: Discriminant

Q44 (Asked in 2011):

Solve x² – 2x – 3 = 0 by completing the square.

Solution: x² – 2x = 3. Add (2/2)² = 1: (x – 1)² = 4. x – 1 = ±2. x = 3, x = -1.

Formula Used: Completing the Square

Q45 (Asked in 2010):

The sum of squares of two consecutive even numbers is 340. Find the numbers.

Solution: x² + (x + 2)² = 340. 2x² + 4x + 4 = 340. x² + 2x – 168 = 0. x = 12, x = -14. Numbers: 12, 14 or -14, -12.

Formula Used: Factorization

Q46 (Asked in 2024):

Solve x² – 10x + 25 = 0 by factorization.

Solution: x² – 10x + 25 = (x – 5)² = 0. Root: x = 5.

Formula Used: Factorization

Q47 (Asked in 2023):

Solve 3x² – 2x – 5 = 0 by quadratic formula.

Solution: a = 3, b = -2, c = -5. D = 4 + 60 = 64. x = [2 ± 8]/6. x = 5/3, x = -1.

Formula Used: Quadratic Formula

Q48 (Asked in 2022):

Find the discriminant of 2x² – 4x + 3 = 0.

Solution: D = (-4)² – 4(2)(3) = 16 – 24 = -8. No real roots.

Formula Used: Discriminant

Q49 (Asked in 2021):

Solve x² + x – 12 = 0 by factorization.

Solution: x² + x – 12 = (x + 4)(x – 3) = 0. Roots: x = -4, x = 3.

Formula Used: Factorization

Q50 (Asked in 2020):

A rectangle’s length is 2 cm more than its width, and its area is 120 cm². Find the dimensions.

Solution: Let width = x. Length = x + 2. x(x + 2) = 120. x² + 2x – 120 = 0. x = 10, x = -12. Width = 10 cm, length = 12 cm.

Formula Used: Factorization

Q51 (Asked in 2019):

Solve 2x² + x – 3 = 0 by completing the square.

Solution: Divide by 2: x² + x/2 = 3/2. Add (1/4)² = 1/16: (x + 1/4)² = 25/16. x = 1, x = -3/2.

Formula Used: Completing the Square

Q52 (Asked in 2018):

Solve x² – 3x – 4 = 0 by factorization.

Solution: x² – 3x – 4 = (x – 4)(x + 1) = 0. Roots: x = 4, x = -1.

Formula Used: Factorization

Q53 (Asked in 2017):

Find the nature of roots of x² + 3x + 3 = 0.

Solution: D = 3² – 4(1)(3) = 9 – 12 = -3. No real roots.

Formula Used: Discriminant

Q54 (Asked in 2016):

Solve 2x² – 5x + 3 = 0 by quadratic formula.

Solution: a = 2, b = -5, c = 3. D = 25 – 24 = 1. x = [5 ± 1]/4. x = 3/2, x = 1.

Formula Used: Quadratic Formula

Q55 (Asked in 2015):

The product of two consecutive numbers is 156. Find the numbers.

Solution: x(x + 1) = 156. x² + x – 156 = 0. Factorize: (x + 13)(x – 12) = 0. x = 12. Numbers: 12, 13.

Formula Used: Factorization

Q56 (Asked in 2014):

Solve x² + 7x + 12 = 0 by factorization.

Solution: x² + 7x + 12 = (x + 4)(x + 3) = 0. Roots: x = -4, x = -3.

Formula Used: Factorization

Q57 (Asked in 2013):

Solve 3x² – x – 4 = 0 by quadratic formula.

Solution: a = 3, b = -1, c = -4. D = 1 + 48 = 49. x = [1 ± 7]/6. x = 4/3, x = -1.

Formula Used: Quadratic Formula

Q58 (Asked in 2012):

Find the discriminant of x² – 10x + 25 = 0.

Solution: D = (-10)² – 4(1)(25) = 100 – 100 = 0. Equal real roots.

Formula Used: Discriminant

Q59 (Asked in 2011):

Solve x² – 5x + 6 = 0 by completing the square.

Solution: x² – 5x = -6. Add (5/2)² = 25/4: (x – 5/2)² = 1/4. x = 3, x = 2.

Formula Used: Completing the Square

Q60 (Asked in 2010):

The area of a rectangle is 96 cm², and its length is 4 cm more than its width. Find the dimensions.

Solution: Let width = x. Length = x + 4. x(x + 4) = 96. x² + 4x – 96 = 0. x = 8, x = -12. Width = 8 cm, length = 12 cm.

Formula Used: Factorization

Q61 (Asked in 2024):

Solve x² – 4x – 12 = 0 by factorization.

Solution: x² – 4x – 12 = (x – 6)(x + 2) = 0. Roots: x = 6, x = -2.

Formula Used: Factorization

Q62 (Asked in 2023):

Solve 2x² + 7x + 6 = 0 by quadratic formula.

Solution: a = 2, b = 7, c = 6. D = 49 – 48 = 1. x = [-7 ± 1]/4. x = -3/2, x = -2.

Formula Used: Quadratic Formula

Q63 (Asked in 2022):

Find the nature of roots of x² + 5x + 7 = 0.

Solution: D = 5² – 4(1)(7) = 25 – 28 = -3. No real roots.

Formula Used: Discriminant

Q64 (Asked in 2021):

Solve x² + 2x – 15 = 0 by factorization.

Solution: x² + 2x – 15 = (x + 5)(x – 3) = 0. Roots: x = -5, x = 3.

Formula Used: Factorization

Q65 (Asked in 2020):

The sum of a number and its reciprocal is 17/4. Find the number.

Solution: x + 1/x = 17/4. x² + 1 = 17x/4. 4x² – 17x + 4 = 0. x = 4, x = 1/4.

Formula Used: Quadratic Formula

Q66 (Asked in 2019):

Solve 3x² + 4x – 4 = 0 by completing the square.

Solution: Divide by 3: x² + 4/3x = 4/3. Add (2/3)² = 4/9: (x + 2/3)² = 16/9. x = 2/3, x = -2.

Formula Used: Completing the Square

Q67 (Asked in 2018):

Solve x² – 9x + 18 = 0 by factorization.

Solution: x² – 9x + 18 = (x – 3)(x – 6) = 0. Roots: x = 3, x = 6.

Formula Used: Factorization

Q68 (Asked in 2017):

Find the discriminant of 2x² – 3x + 2 = 0.

Solution: D = (-3)² – 4(2)(2) = 9 – 16 = -7. No real roots.

Formula Used: Discriminant

Q69 (Asked in 2016):

Solve 2x² – x – 1 = 0 by quadratic formula.

Solution: a = 2, b = -1, c = -1. D = 1 + 8 = 9. x = [1 ± 3]/4. x = 1, x = -1/2.

Formula Used: Quadratic Formula

Q70 (Asked in 2015):

The product of two consecutive even numbers is 624. Find the numbers.

Solution: x(x + 2) = 624. x² + 2x – 624 = 0. Factorize: (x + 26)(x – 24) = 0. x = 24. Numbers: 24, 26.

Formula Used: Factorization

Q71 (Asked in 2014):

Solve x² + 8x + 15 = 0 by factorization.

Solution: x² + 8x + 15 = (x + 5)(x + 3) = 0. Roots: x = -5, x = -3.

Formula Used: Factorization

Q72 (Asked in 2013):

Solve 3x² + 2x – 8 = 0 by quadratic formula.

Solution: a = 3, b = 2, c = -8. D = 4 + 96 = 100. x = [-2 ± 10]/6. x = 4/3, x = -2.

Formula Used: Quadratic Formula

Q73 (Asked in 2012):

Find the nature of roots of x² – 12x + 36 = 0.

Solution: D = (-12)² – 4(1)(36) = 144 – 144 = 0. Equal real roots.

Formula Used: Discriminant

Q74 (Asked in 2011):

Solve x² – 3x – 10 = 0 by completing the square.

Solution: x² – 3x = 10. Add (3/2)² = 9/4: (x – 3/2)² = 49/4. x = 5, x = -2.

Formula Used: Completing the Square

Q75 (Asked in 2010):

The sum of squares of two consecutive numbers is 265. Find the numbers.

Solution: x² + (x + 1)² = 265. 2x² + 2x + 1 = 265. x² + x – 132 = 0. x = 11, x = -12. Numbers: 11, 12 or -12, -11.

Formula Used: Factorization

Q76 (Asked in 2024):

Solve x² – 7x + 10 = 0 by factorization.

Solution: x² – 7x + 10 = (x – 2)(x – 5) = 0. Roots: x = 2, x = 5.

Formula Used: Factorization

Q77 (Asked in 2023):

Solve 2x² + 4x – 6 = 0 by quadratic formula.

Solution: a = 2, b = 4, c = -6. D = 16 + 48 = 64. x = [-4 ± 8]/4. x = 1, x = -3.

Formula Used: Quadratic Formula

Q78 (Asked in 2022):

Find the discriminant of x² + 4x + 5 = 0.

Solution: D = 4² – 4(1)(5) = 16 – 20 = -4. No real roots.

Formula Used: Discriminant

Q79 (Asked in 2021):

Solve x² + 4x – 5 = 0 by factorization.

Solution: x² + 4x – 5 = (x + 5)(x – 1) = 0. Roots: x = -5, x = 1.

Formula Used: Factorization

Q80 (Asked in 2020):

A train travels 300 km at a uniform speed. If the speed had been 10 km/h more, it would have taken 1 hour less. Find the speed.

Solution: Let speed = x km/h. 300/x – 300/(x + 10) = 1. x² + 10x – 3000 = 0. x = 50. Speed: 50 km/h.

Formula Used: Quadratic Formula

Q81 (Asked in 2019):

Solve 2x² – 3x – 5 = 0 by completing the square.

Solution: Divide by 2: x² – 3/2x = 5/2. Add (3/4)² = 9/16: (x – 3/4)² = 49/16. x = 5/2, x = -1.

Formula Used: Completing the Square

Q82 (Asked in 2018):

Solve x² – 6x + 8 = 0 by factorization.

Solution: x² – 6x + 8 = (x – 4)(x – 2) = 0. Roots: x = 4, x = 2.

Formula Used: Factorization

Q83 (Asked in 2017):

Find the discriminant of x² + 2x + 2 = 0.

Solution: D = 2² – 4(1)(2) = 4 – 8 = -4. No real roots.

Formula Used: Discriminant

Q84 (Asked in 2016):

Solve 3x² – 5x – 2 = 0 by quadratic formula.

Solution: a = 3, b = -5, c = -2. D = 25 + 24 = 49. x = [5 ± 7]/6. x = 2, x = -1/3.

Formula Used: Quadratic Formula

Q85 (Asked in 2015):

The product of two consecutive odd numbers is 399. Find the numbers.

Solution: x(x + 2) = 399. x² + 2x – 399 = 0. Factorize: (x + 21)(x – 19) = 0. x = 19. Numbers: 19, 21.

Formula Used: Factorization

Q86 (Asked in 2014):

Solve x² + 3x – 4 = 0 by factorization.

Solution: x² + 3x – 4 = (x + 4)(x – 1) = 0. Roots: x = -4, x = 1.

Formula Used: Factorization

Q87 (Asked in 2013):

Solve 2x² + x – 4 = 0 by quadratic formula.

Solution: a = 2, b = 1, c = -4. D = 1 + 32 = 33. x = [-1 ± √33]/4.

Formula Used: Quadratic Formula

Q88 (Asked in 2012):

Find the nature of roots of x² – 4x + 5 = 0.

Solution: D = (-4)² – 4(1)(5) = 16 – 20 = -4. No real roots.

Formula Used: Discriminant

Q89 (Asked in 2011):

Solve x² – x – 12 = 0 by completing the square.

Solution: x² – x = 12. Add (1/2)² = 1/4: (x – 1/2)² = 49/4. x = 4, x = -3.

Formula Used: Completing the Square

Q90 (Asked in 2010):

The area of a rectangle is 80 cm², and its length is 6 cm more than its width. Find the dimensions.

Solution: Let width = x. Length = x + 6. x(x + 6) = 80. x² + 6x – 80 = 0. x = 8, x = -10. Width = 8 cm, length = 14 cm.

Formula Used: Factorization

Q91 (Asked in 2024):

Solve x² – 8x + 12 = 0 by factorization.

Solution: x² – 8x + 12 = (x – 6)(x – 2) = 0. Roots: x = 6, x = 2.

Formula Used: Factorization

Q92 (Asked in 2023):

Solve 2x² – 5x – 3 = 0 by quadratic formula.

Solution: a = 2, b = -5, c = -3. D = 25 + 24 = 49. x = [5 ± 7]/4. x = 3, x = -1/2.

Formula Used: Quadratic Formula

Q93 (Asked in 2022):

Find the nature of roots of x² + 6x + 10 = 0.

Solution: D = 6² – 4(1)(10) = 36 – 40 = -4. No real roots.

Formula Used: Discriminant

Q94 (Asked in 2021):

Solve x² + 5x – 6 = 0 by factorization.

Solution: x² + 5x – 6 = (x + 6)(x – 1) = 0. Roots: x = -6, x = 1.

Formula Used: Factorization

Q95 (Asked in 2020):

The sum of a number and its reciprocal is 13/6. Find the number.

Solution: x + 1/x = 13/6. 6x² – 13x + 6 = 0. x = 3/2, x = 2/3.

Formula Used: Quadratic Formula

Q96 (Asked in 2019):

Solve 2x² + x – 1 = 0 by completing the square.

Solution: Divide by 2: x² + x/2 = 1/2. Add (1/4)² = 1/16: (x + 1/4)² = 9/16. x = 1/2, x = -1.

Formula Used: Completing the Square

Q97 (Asked in 2018):

Solve x² – 7x + 12 = 0 by factorization.

Solution: x² – 7x + 12 = (x – 3)(x – 4) = 0. Roots: x = 3, x = 4.

Formula Used: Factorization

Q98 (Asked in 2017):

Find the discriminant of x² – 2x + 3 = 0.

Solution: D = (-2)² – 4(1)(3) = 4 – 12 = -8. No real roots.

Formula Used: Discriminant

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