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About Quadratic Equations — Class 10 CBSE

Solve quadratic equations using factorization and quadratic formula; determine nature of roots. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 4). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Quadratic Equations — solved examples for Class 10 CBSE

Example 1easy

Which of the following expressions, when simplified, results in a quadratic equation?
  1. A)(x + 1)² = 2x + 3
  2. B)x(x + 2) = x² + 5
  3. C)x³ - 4x² + 5 = 0
  4. D)(x - 2)(x + 2) = x² - 4

Step-by-step solution

  1. Option A: (x + 1)² = 2x + 3. Expanding the left side gives x² + 2x + 1 = 2x + 3. Simplifying, we get x² - 2 = 0, which is a quadratic equation (a=1, b=0, c=-2).
  2. Option B: x(x + 2) = x² + 5. Expanding gives x² + 2x = x² + 5. Simplifying, we get 2x - 5 = 0, which is a linear equation.
  3. Option C: x³ - 4x² + 5 = 0. This is a cubic equation, not quadratic, as the highest power of x is 3.
  4. Option D: (x - 2)(x + 2) = x² - 4. Expanding gives x² - 4 = x² - 4. Simplifying, we get 0 = 0, which is an identity and not an equation in x.

Answer: (x + 1)² = 2x + 3

Example 2medium

Which of the following equations is NOT a quadratic equation?
  1. A)(x - 2)² + 1 = 2x - 3
  2. B)x(x + 1) + 8 = (x + 2)(x - 2)
  3. C)x(2x + 3) = x² + 1
  4. D)(x + 2)³ = x³ - 4

Step-by-step solution

  1. Let's simplify each option:
  2. A) (x - 2)² + 1 = 2x - 3 => x² - 4x + 4 + 1 = 2x - 3 => x² - 6x + 8 = 0. This is a quadratic equation.
  3. B) x(x + 1) + 8 = (x + 2)(x - 2) => x² + x + 8 = x² - 4 => x + 12 = 0. This is a linear equation, not quadratic.
  4. C) x(2x + 3) = x² + 1 => 2x² + 3x = x² + 1 => x² + 3x - 1 = 0. This is a quadratic equation.
  5. D) (x + 2)³ = x³ - 4 => x³ + 6x² + 12x + 8 = x³ - 4 => 6x² + 12x + 12 = 0. This is a quadratic equation.

Answer: x(x + 1) + 8 = (x + 2)(x - 2)

Example 3hard

Consider the quadratic equation ax² + bx + c = 0, where a, b, c are rational numbers and a ≠ 0. If the roots of the equation are rational, which of the following statements MUST be true about its discriminant, D = b² - 4ac?
  1. A)D must be positive.
  2. B)D must be a perfect square of a rational number.
  3. C)D must be a perfect square of an integer.
  4. D)D must be non-negative.

Step-by-step solution

  1. For the roots x = [-b ± √(D)] / 2a to be rational, the term √(D) must be rational.
  2. For √(D) to be rational, D must be a perfect square of a rational number. If D is a perfect square of a rational number, say D = (p/q)², then √(D) = p/q, which is rational.
  3. Since a, b, c are rational, -b and 2a are rational. Therefore, if √(D) is rational, the roots will be rational.
  4. Option C (perfect square of an integer) is too restrictive, as D could be (1/4) or (9/16), etc., which are perfect squares of rational numbers but not integers. Option D (non-negative) only ensures real roots, not necessarily rational.

Answer: D must be a perfect square of a rational number.

Practice questions on Quadratic Equations

  1. Q1.easy

    If 'p' is a root of the quadratic equation ax² + bx + c = 0, which of the following statements must be true?
    1. A)a + b + c = 0
    2. B)ap² + bp + c = 0
    3. C)p = -b / 2a
    4. D)b² - 4ac > 0
    Show answer

    Answer: ap² + bp + c = 0

    Hint: Remember the definition of a root: it is a value that satisfies the given equation when substituted for the variable.

  2. Q2.easy

    Ravi is trying to solve the quadratic equation x² - 7x + 12 = 0 by factorization. He writes the first step as x² - 3x - 4x + 12 = 0. Which of the following statements about Ravi's first step is true?
    1. A)Ravi has made a mistake in splitting the middle term.
    2. B)Ravi has correctly split the middle term, and the next step is x(x-3) - 4(x-3) = 0.
    3. C)Ravi should have split the middle term as -2x - 5x.
    4. D)Ravi should have used the quadratic formula instead of factorization.
    Show answer

    Answer: Ravi has correctly split the middle term, and the next step is x(x-3) - 4(x-3) = 0.

    Hint: To factorize x² + bx + c, you need to find two numbers whose sum is 'b' and product is 'c'. Check if Ravi's split satisfies these conditions.

  3. Q3.easy

    The product of two consecutive positive integers is 306. Which of the following quadratic equations represents this situation?
    1. A)x² + x + 306 = 0
    2. B)x² - x + 306 = 0
    3. C)x² + x - 306 = 0
    4. D)x² - x - 306 = 0
    Show answer

    Answer: x² + x - 306 = 0

    Hint: Let the first positive integer be 'x'. How would you represent the next consecutive positive integer? Then, form an equation based on their product.

  4. Q4.medium

    Find the roots of the quadratic equation 6x² + 17x + 5 = 0 by factorization.
    1. A)x = -1/3, x = -5/2
    2. B)x = 1/3, x = 5/2
    3. C)x = -1/2, x = -5/3
    4. D)x = 1/2, x = 5/3
    Show answer

    Answer: x = -1/3, x = -5/2

    Hint: To factorize, split the middle term (17x) into two terms such that their product equals the product of the first and last terms (6x² × 5).

  5. Q5.medium

    For what value of p does the quadratic equation px(x - 2) + 6 = 0 have two equal real roots?
    1. A)p = 3
    2. B)p = 6
    3. C)p = -3
    4. D)p = 9
    Show answer

    Answer: p = 6

    Hint: First, rewrite the equation in the standard form ax² + bx + c = 0. For equal real roots, the discriminant (D = b² - 4ac) must be zero.

  6. Q6.medium

    The product of two consecutive positive even integers is 288. Find the integers.
    1. A)12 and 14
    2. B)16 and 18
    3. C)18 and 20
    4. D)10 and 12
    Show answer

    Answer: 16 and 18

    Hint: Let the first even integer be x. What would be the next consecutive even integer? Set up an equation for their product.

  7. Q7.hard

    For what integer values of 'k' will the quadratic equation (k+1)x² - 2(k-1)x + 1 = 0 have equal roots?
    1. A)k = 0 only
    2. B)k = 0 or k = 2
    3. C)k = 2 only
    4. D)k = 0 or k = -2
    Show answer

    Answer: k = 0 or k = 2

    Hint: For a quadratic equation to have equal roots, its discriminant must be zero. Also, remember that the coefficient of x² cannot be zero for it to be a quadratic equation.

  8. Q8.hard

    A rectangular park is to be designed whose breadth is 3 meters less than its length. Its area is to be 4 square meters more than the area of a triangular park with base equal to the length of the rectangular park and height 12 meters. Find the length of the rectangular park.
    1. A)7 meters
    2. B)10 meters
    3. C)12 meters
    4. D)15 meters
    Show answer

    Answer: 10 meters

    Hint: Formulate expressions for the areas of both the rectangular and triangular parks in terms of the length of the rectangular park. Then set up the equation based on the given condition.

  9. Q9.hard

    A motorboat whose speed in still water is 18 km/h takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
    1. A)4 km/h
    2. B)6 km/h
    3. C)8 km/h
    4. D)10 km/h
    Show answer

    Answer: 6 km/h

    Hint: Let the speed of the stream be 'x'. Express the speeds upstream and downstream, and then use the formula Time = Distance / Speed to set up an equation involving the time difference.

These are 9 of the 60 questions available for Quadratic Equations. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.