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Class 10 Maths Chapter 4: Quadratic Equations — Important Questions & Sample Paper

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Reviewed by qpaper's CBSE curriculum team · Edited by Mohit · Updated June 2026

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Yes — this page has 44+ original Class 10 Mathematics Chapter 4 (“Quadratic Equations”) important questions with answers (Multiple Choice (MCQ), Assertion–Reason, Short Answer, Short Answer, Long Answer, Case Study). Practise them free, or generate a full CBSE board-pattern sample paper (80 marks) and export it to PDF or Word — in English & Hindi, for 2026-27.

Chapter 4 of Class 10 Maths, Quadratic Equations, introduces the standard form ax²+bx+c=0 (a ≠ 0) and teaches how to identify quadratic equations. Students learn solution methods: factorization (splitting the middle term), completing the square, and the quadratic formula x = [-b ± √(b²-4ac)] / 2a. The discriminant D = b²-4ac decides the nature of roots: D > 0 (two distinct real roots), D = 0 (two equal real roots), D < 0 (no real roots). The chapter also covers reducing equations to quadratic form—like cross-multiplying rational expressions or expanding squared binomials—before solving. Exam questions commonly ask to check if an equation is quadratic, find unknown coefficients (k) given root conditions (equal roots, a known root), solve using the formula, and tackle word problems. Typical word problems involve speed-distance-time (e.g., train speed and time difference) and geometry (combined area of squares, perimeter relations). Students must set up equations from real-life situations, simplify, and interpret the discriminant. Our question bank mirrors this variety: questions like finding k for equal roots in x²+2x+k=0, solving (x/(x+1)) + ((x+1)/x) = 34/15, determining the nature of roots of 3x²-4√3 x+4=0, and word problems on train speed and areas of squares. Mastery of quadratic equations builds a strong foundation for higher algebra and is crucial for the CBSE board exams.

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Mathematics — Quadratic Equations

Class 10Time: 3 hrsMax Marks: 80

SECTION A

  1. 1.

    For what value of p does the equation x^2 - 4x + p = 0 have real and distinct roots?

    (a) p < 4(b) p > 4(c) p = 4(d) p ≥ 4
    1
  2. 2.

    The roots of the equation x^2 - 5x + 6 = 0 are:

    (a) 2 and 3(b) -2 and -3(c) 2 and -3(d) -2 and 3
    1
  3. 3.

    The discriminant of the quadratic equation 2x^2 - 4x + 3 = 0 is:

    (a) -8(b) 8(c) -4(d) 4
    1

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Marks distribution & blueprint

In a CBSE exam, this chapter typically contributes questions across the following types. The last column shows how many original questions of each type we have ready in our bank for this chapter:

Question typeMarks eachIn our bank
Multiple Choice (MCQ)1 mark13
Assertion–Reason1 mark6
Short Answer2 marks8
Short Answer3 marks6
Long Answer5 marks5
Case Study4 marks6

44 original, exam-style questions in our bank for this chapter — with answers.

Important & sample questions (with answers)

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  1. Q1. For what value of p does the equation x^2 - 4x + p = 0 have real and distinct roots?

    1 mark
    Multiple Choice (MCQ)
    (A) p < 4(B) p > 4(C) p = 4(D) p ≥ 4
    ▸ Answer

    p < 4

  2. Q2. The roots of the equation x^2 - 5x + 6 = 0 are:

    1 mark
    Multiple Choice (MCQ)
    (A) 2 and 3(B) -2 and -3(C) 2 and -3(D) -2 and 3
    ▸ Answer

    2 and 3

  3. Q3. The discriminant of the quadratic equation 2x^2 - 4x + 3 = 0 is:

    1 mark
    Multiple Choice (MCQ)
    (A) -8(B) 8(C) -4(D) 4
    ▸ Answer

    -8

  4. Q4. The nature of the roots of the quadratic equation 2x^2 - √5 x + 1 = 0 is:

    1 mark
    Multiple Choice (MCQ)
    (A) two distinct real roots(B) two equal real roots(C) no real roots(D) cannot be determined
    ▸ Answer

    no real roots

  5. Q5. Assertion (A): The quadratic equation x^2 - 7x + 12 = 0 has two distinct positive roots. Reason (R): For a quadratic equation to have real roots, the discriminant must be a perfect square.

    1 mark
    Assertion–Reason
    (A) Both A and R are true and R is the correct explanation of A.(B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
    ▸ Answer

    A is true but R is false.

  6. Q6. Find the roots of the quadratic equation √2 x^2 + 7x + 5√2 = 0 by factorization.

    2 marks
    Short Answer
    ▸ Answer

    -5/√2 and -√2

  7. Q7. Solve for x: 1/(x+4) - 1/(x-7) = 11/30, x ≠ -4, 7.

    2 marks
    Short Answer
    ▸ Answer

    x = 1 or x = 2

  8. Q8. A motorboat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream. Find the speed of the stream.

    3 marks
    Short Answer
    ▸ Answer

    6 km/h

  9. Q9. Find the discriminant of the quadratic equation 2x^2 - 4x + 3 = 0 and hence state the nature of its roots.

    3 marks
    Short Answer
    ▸ Answer

    Discriminant = -8, no real roots.

  10. Q10. A motorboat whose speed in still water is 24 km/h takes 1 hour more to go 32 km upstream than to return downstream to the same spot. Find the speed of the stream.

    5 marks
    Long Answer
    ▸ Answer

    The speed of the stream is 8 km/h.

  11. Q11. The sum of the areas of two squares is 544 m², and the difference of their perimeters is 32 m. Find the sides of the two squares.

    5 marks
    Long Answer
    ▸ Answer

    The sides of the squares are 20 m and 12 m.

  12. Q12. A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot.

    4 marks
    Case Study
    1. (i) If the speed of the stream is x km/h, write an equation for the difference in the times taken for upstream and downstream journeys.1 mark
    2. (ii) Simplify the equation to obtain a quadratic equation in x.1 mark
    3. (iii) Find the speed of the stream.2 marks
    ▸ Answer

    Speed of stream = 6 km/h

Frequently asked questions

What is the discriminant of a quadratic equation and how does it help determine the nature of roots?

The discriminant D = b²-4ac from the quadratic formula ax²+bx+c=0. If D > 0, the equation has two distinct real roots; if D = 0, it has two equal real roots; if D < 0, it has no real roots. This helps quickly predict root nature without solving, often tested with questions like 'Find k for equal roots' or 'Check the nature of roots of 3x²-4√3 x+4=0'.

What are the key steps to solve a word problem that leads to a quadratic equation?

First, identify the unknown and denote it by a variable. Translate the conditions into an equation using relationships (time = distance/speed, area relations). Simplify into ax²+bx+c=0. Solve using factorization or quadratic formula. Check validity in context (e.g., non-negative speed). Example: a train problem where increased speed reduces travel time by 2 hours.

How do I decide which method—factorization, completing the square, or quadratic formula—to use for solving?

Use factorization when the middle term splits easily. Completing the square is handy when a=1 and the coefficient of x is even. The quadratic formula works universally, making it reliable for irrational roots or complex coefficients. In exams, you may choose any method, but the formula is safest, especially when exact decimal roots are required.

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