Chapter 4: Quadratic Equations

VVI questions — the ones the board asked most (20)

1. Q1Short answer2 marksAsked: 2022, 2025

   Find two consecutive positive integers, sum of whose squares is 365.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 75 s

   Model answer · quadratic equations, word problem (consecutive integers)

   1. 1.Let the two consecutive positive integers be x and x + 1.0.5 marks
   2. 2.According to the question, the sum of their squares is 365. So, the equation is x² + (x+1)² = 365.0.5 marks
   3. 3.Solving the equation, x² + x² + 2x + 1 = 365, which gives 2x² + 2x - 364 = 0, i.e., x² + x - 182 = 0, a quadratic equation in x.0.5 marks
   4. 4.Factorising x² + x - 182 = 0, we get (x - 13)(x + 14) = 0, so x = 13 or x = -14. Since x is given to be a positive integer, x ≠ -14, so x = 13. Thus, the two consecutive positive integers are 13 and 14.0.5 marks

   • Write 'Let the two consecutive positive integers be x and x + 1' as the first line.
   • State the equation x² + (x+1)² = 365 before simplifying it.
   • End by rejecting the negative root and writing both positive integers, 13 and 14.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

2. Q2Short answer2 marksAsked: 2020, 2022

   Find the nature of the roots of the quadratic equation 4x^2 - 5x + 3 = 0.

   

   Medha Ma’am

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   Hear it from Medha Ma’am · 55 s

   Model answer · nature of roots, quadratic equation

   1. 1.Discriminant D = b² − 4ac; with a = 4, b = −5, c = 3, D = (−5)² − 4 × 4 × 3 = 25 − 48 = −23.1 mark
   2. 2.Since D < 0, the equation has no real roots.1 mark

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

3. Q3Short answer2 marksAsked: 2020, 2022

   If the sum of two natural numbers is 27 and their product is 182, then find the numbers.

   

   Medha Ma’am

   AskMedha’s AI teacher

   Hear it from Medha Ma’am · 85 s

   Model answer · word problem, quadratic equations

   1. 1.Let the two numbers be x and (27 - x), so that x + (27 - x) = 27.0.5 marks
   2. 2.Given, the product of the two numbers is 182, so x(27 - x) = 182. On solving, 27x - x^2 = 182, which gives x^2 - 27x + 182 = 0.0.5 marks
   3. 3.By factorisation method, x^2 - 27x + 182 = 0 can be written as x^2 - 13x - 14x + 182 = 0, i.e., x(x - 13) - 14(x - 13) = 0, so (x - 13)(x - 14) = 0. This gives x = 13 or x = 14.0.5 marks
   4. 4.Thus, the required numbers are 13 and 14, since 13 + 14 = 27 and 13 × 14 = 182.0.5 marks

   • Write the equation x + (27-x) = 27 before forming the quadratic.
   • Show the working step x(27-x) = 182 leading to x^2 - 27x + 182 = 0.
   • State both roots 13 and 14 clearly as the final answer.

   Marks split from a structure derived from past papers (checked twice); not Bihar Board’s official scheme

4. Q4Short answer2 marksAsked: 2017, 2022

   Solve sqrt(13 - x^2) = x + 5.

   How the model answer begins

   1. Squaring both sides: 13 − x² = (x + 5)² = x² + 10x + 25.

   The full answer from Medha Ma’am →
5. Q5Long answer5 marksAsked: 2026

   Quadratic equation ax^2 + bx + c = 0 where a, b & c are real numbers and a ≠ 0 then prove that x = (-b ± sqrt(b^2 - 4ac)) / 2a.

   How the model answer begins

   1. The given quadratic equation is ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Dividing throughout by a, we get x² + (b/a)x + (c/a) = 0.

   The full answer from Medha Ma’am →
6. Q6Long answer5 marksAsked: 2022

   The difference of the squares of two numbers is 180. If the square of the smaller number is eight times the larger number, find the two numbers.

   How the model answer begins

   1. Let the larger number be x and the smaller number be y.

   The full answer from Medha Ma’am →
7. Q7Long answer5 marksAsked: 2022

   The perpendicular of a right angled triangle is 7 cm less than its base and its hypotenuse is 13 cm. Find the base of the triangle.

   How the model answer begins

   1. Let the base of the right angled triangle be x cm. Since the perpendicular (altitude) is 7 cm less than its base, the perpendicular will be (x - 7) cm.

   The full answer from Medha Ma’am →
8. Q8Long answer5 marksAsked: 2023

   Divide 50 into two parts so that the sum of their reciprocals is 1/12.

   How the model answer begins

   1. Let the first part be x. Then the second part, since their sum is 50, is 50 - x.

   The full answer from Medha Ma’am →
9. Q9Long answer5 marksAsked: 2023

   Divide 50 into two parts so that the sum of their reciprocals is 1/12.

   How the model answer begins

   1. Let the first part be x. Since the two parts together make 50, the second part will be 50 - x.

   The full answer from Medha Ma’am →
10. Q10Long answer5 marksAsked: 2021

    A two-digit number is four times the sum of its digits and twice the product of its digits, find the number.

    How the model answer begins

    1. Let the tens digit of the required number be x and the units digit be y. Then the number is 10x + y.

    The full answer from Medha Ma’am →
11. Q11Long answer5 marksAsked: 2021

    The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was 124. Find their present ages.

    How the model answer begins

    1. Let the father's present age be x years. Since the sum of the ages of the father and his son is 45 years, the son's present age = (45 - x) years.

    The full answer from Medha Ma’am →
12. Q12Long answer5 marksAsked: 2021

    A motorboat, whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to go the same distance downstream. Find the speed of the stream.

    How the model answer begins

    1. Let the speed of the stream be x km/h; then the upstream speed is (18 − x) km/h and the downstream speed (18 + x) km/h.

    The full answer from Medha Ma’am →
13. Q13Short answer2 marksAsked: 2025

    Find the discriminant of the quadratic equation 2x^2 + 5x - 3 = 0 and find the nature of the roots also.

    How the model answer begins

    1. Comparing the quadratic equation 2x^2 + 5x - 3 = 0 with ax^2 + bx + c = 0, we get a = 2, b = 5, c = -3. The discriminant of this quadratic equation is given by D = b^2 - 4ac.

    The full answer from Medha Ma’am →
14. Q14Short answer2 marksAsked: 2025

    Using quadratic formula find the roots of the equation 2x^2 - 2sqrt(2)x + 1 = 0.

    How the model answer begins

    1. Comparing the given equation 2x^2 - 2\sqrt2 x + 1 = 0 with ax^2 + bx + c = 0, we get a = 2, b = -2\sqrt2 and c = 1.

    The full answer from Medha Ma’am →
15. Q15Short answer2 marksAsked: 2025

    The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Write the equation for this statement.

    How the model answer begins

    1. Let the larger number be x and the smaller y; the difference of their squares is 180, so x² − y² = 180.

    The full answer from Medha Ma’am →
16. Q16Short answer2 marksAsked: 2026

    Find the sum and product of the roots of the quadratic equation x^2 - x + 3 = 0.

    How the model answer begins

    1. Comparing the quadratic equation x^2 - x + 3 = 0 with ax^2 + bx + c = 0, we get a = 1, b = -1, c = 3.

    The full answer from Medha Ma’am →
17. Q17Short answer2 marksAsked: 2026

    Find the discriminant of the quadratic equation 3y^2 - 2sqrt(6)y + 2 = 0 and find the nature of the roots also.

    How the model answer begins

    1. Comparing the given quadratic equation 3y^2 - 2√6y + 2 = 0 with the standard form ay^2 + by + c = 0, we get a = 3, b = -2√6 and c = 2.

    The full answer from Medha Ma’am →
18. Q18Short answer2 marksAsked: 2026

    Using quadratic formula find the roots of the equation 6x^2 + 5x - 6 = 0.

    How the model answer begins

    1. Comparing the equation 6x^2 + 5x - 6 = 0 with ax^2 + bx + c = 0, we get a = 6, b = 5, c = -6. We will use the quadratic formula to find the roots.

    The full answer from Medha Ma’am →
19. Q19Short answer2 marksAsked: 2022

    Find the roots of the quadratic equation 2x^2 - 5x + 3 = 0 using formula.

    How the model answer begins

    1. Quadratic formula x = [−b ± √(b² − 4ac)]/2a; in 2x² − 5x + 3 = 0, a = 2, b = −5, c = 3.

    The full answer from Medha Ma’am →
20. Q20Short answer2 marksAsked: 2022

    For what values of k are the roots of the quadratic equation x^2 - 4kx + k = 0 equal?

    How the model answer begins

    1. Comparing the given quadratic equation x^2 - 4kx + k = 0 with the standard form ax^2 + bx + c = 0, we get a = 1, b = -4k, c = k.

    The full answer from Medha Ma’am →

23 more questions of this chapter, each with its model answer, in the AskMedha app.

Objective questions with the board’s key (10)

1. 1. Bihar Board 2026

   Nature of the roots of the quadratic equation x^2 - 3x - 10 = 0 is

   • (A) real and unequal ✓
   • (B) real and equal
   • (C) non-real
   • (D) none of these

   Answer: (A)

2. 2. Bihar Board 2026

   The value of the discriminant of the quadratic equation bx^2 + ax + c = 0, where b != 0 is

   • (A) a^2 - 4bc ✓
   • (B) b^2 - 4ac
   • (C) c^2 - 4ab
   • (D) b^2 + 4ac

   Answer: (A)

3. 3. Bihar Board 2026

   ax^2 + bx + c = 0 is said to be a quadratic equation in variable x where a, b & c are real numbers then

   • (A) a != 0 ✓
   • (B) b != 0
   • (C) c != 0
   • (D) a = 0

   Answer: (A)

4. 4. Bihar Board 2026

   Discriminant of the quadratic equation 5x^2 - 6x - 2 = 0 is

   • (A) 66
   • (B) 76 ✓
   • (C) 73
   • (D) 72

   Answer: (B)

5. 5. Bihar Board 2026

   The discriminant of the quadratic equation x^2 - 3x + 2 = 0 is

   • (A) 17
   • (B) 27
   • (C) 1 ✓
   • (D) 5

   Answer: (C)

6. 6. Bihar Board 2026

   The roots of the equation cx^2 + ax + b = 0 will not be real, if

   • (A) c^2 - 4ab < 0
   • (B) b^2 - 4ac < 0
   • (C) a^2 - 4bc < 0 ✓
   • (D) a^2 - 4bc > 0

   Answer: (C)

7. 7. Bihar Board 2026

   The ratio of the sum of roots and product of roots of the equation x^2 - 6x + 4 = 0 is

   • (A) 1 : 4
   • (B) 2 : 3
   • (C) 3 : 2 ✓
   • (D) 1 : 3

   Answer: (C)

8. 8. Bihar Board 2026

   Which of the following is a quadratic equation ?

   • (A) (x+3)^2 = x^2 + 7x
   • (B) (x+2)^2 = 2x + 2 ✓
   • (C) (sqrt(x)+2)^2 = x + 2
   • (D) (2x+7)^2 = 4x^2 + 5

   Answer: (B)

9. 9. Bihar Board 2025

   For what value of k are the roots of the quadratic equation kx^2 - 6x + 1 = 0 real and equal?

   • (A) 6
   • (B) 8
   • (C) 9 ✓
   • (D) 10

   Answer: (C)

10. 10. Bihar Board 2025

    If alpha and beta are roots of the quadratic equation 3x^2 - 5x + 2 = 0 then the value of alpha^2 + beta^2 is

    • (A) 13/9 ✓
    • (B) 9/13
    • (C) 5/3
    • (D) 3/5

    Answer: (A)

Questions parents and students ask

How many marks does "Quadratic Equations" carry in Bihar Board Class 10 Maths?

In the Bihar Board papers of 2021–2026, this chapter carried about 8 marks and about 11 questions a year; by marks it ranks 3 in Maths.

Which are the most important (VVI) questions of "Quadratic Equations"?

The 20 questions above are from the board’s real papers, each with the years it was asked; the most-asked come first with their full model answers. All 43 written questions of this chapter, with answers, are in the AskMedha app.

Do the answers follow the board’s marking scheme?

BSEB does not publish a marking scheme. Each answer is written point by point on a structure derived from past papers and checked by a second, independent model; it is never called the board’s official scheme.