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About Quadratic Equations — Class 10 Olympiad

Solve quadratics by factoring, completing the square, and quadratic formula; analyze discriminant and nature of roots. This topic is part of the Olympiad Class 10 mathematics syllabus (chapter: Module 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 Olympiad

Example 1easy

For what value of 'm' is the equation `(m-3)x^2 + (m^2-9)x + (m+3) = 0` NOT a quadratic equation?
  1. A)A) 0
  2. B)B) -3
  3. C)C) 9
  4. D)D) 3

Step-by-step solution

  1. For an equation to be quadratic, the coefficient of the `x^2` term must be non-zero. That is, `a ≠ 0` in `ax^2 + bx + c = 0`.
  2. In the given equation, the coefficient of `x^2` is `(m-3)`.
  3. If `(m-3) = 0`, then `m = 3`. In this case, the `x^2` term vanishes, and the equation is no longer quadratic.
  4. When `m=3`, the equation becomes `0x^2 + (3^2-9)x + (3+3) = 0`, which simplifies to `0x^2 + 0x + 6 = 0`, or `6 = 0`, which is a false statement. Thus, it's not a quadratic equation.

Answer: D) 3

Example 2medium

For what integer value(s) of 'm' is the equation (m^2 - 9)x^2 + (m+3)x + 7 = 0 NOT a quadratic equation?
  1. A)m = 3 only
  2. B)m = -3 only
  3. C)m = 3 or m = -3
  4. D)No such integer 'm' exists

Step-by-step solution

  1. For an equation to be a quadratic equation, the coefficient of the x^2 term must not be zero.
  2. In the given equation, the coefficient of x^2 is (m^2 - 9).
  3. Setting this to zero: m^2 - 9 = 0.
  4. This factors as (m-3)(m+3) = 0. Thus, m = 3 or m = -3. If m takes either of these values, the x^2 term vanishes, and the equation ceases to be quadratic.

Answer: m = 3 or m = -3

Example 3hard

For what integer value(s) of 'm' does the equation (m² - 9)x² + (m + 3)x + 2 = 0 have exactly one real root?
  1. A)m = 3 only
  2. B)m = -3 only
  3. C)m = 3 or m = 27/7
  4. D)m = 3

Step-by-step solution

  1. Case 1: The equation is linear. This occurs when the coefficient of x² is zero, i.e., m² - 9 = 0. This implies m = 3 or m = -3.
  2. If m = 3, the equation becomes (3² - 9)x² + (3 + 3)x + 2 = 0, which simplifies to 0x² + 6x + 2 = 0, or 6x = -2. This gives x = -1/3, which is exactly one real root. So, m = 3 is a solution.
  3. If m = -3, the equation becomes ((-3)² - 9)x² + (-3 + 3)x + 2 = 0, which simplifies to 0x² + 0x + 2 = 0, or 2 = 0. This is a contradiction, meaning there are no solutions. So, m = -3 is not a solution.
  4. Case 2: The equation is a quadratic with exactly one real root. This means the discriminant (D) must be zero. D = (m+3)² - 4(m²-9)(2) = 0. This simplifies to (m+3)² - 8(m-3)(m+3) = 0. Factoring out (m+3), we get (m+3)[(m+3) - 8(m-3)] = 0, which leads to (m+3)(-7m + 27) = 0. So, m = -3 or m = 27/7. However, for this case, the equation must be quadratic, meaning m²-9 ≠ 0. Therefore, m ≠ -3. So, m = 27/7 is a solution from this case. Since the question asks for integer values, m = 27/7 is not an integer.
  5. Combining both cases, the only integer value of 'm' for which the equation has exactly one real root is m = 3.

Answer: m = 3

Practice questions on Quadratic Equations

  1. Q1.easy

    Find the roots of the equation `x(x+1) + (x+2)(x+3) = 42`.
    1. A)A) 2, -5
    2. B)B) 3, -6
    3. C)C) 4, -7
    4. D)D) 1, -18
    Show answer

    Answer: B) 3, -6

    Hint: First, expand and simplify the equation into the standard quadratic form `ax^2 + bx + c = 0` before attempting to find the roots.

  2. Q2.easy

    If the quadratic equation `(k+1)x^2 - 2(k-1)x + 1 = 0` has real and equal roots, then find the value(s) of `k`.
    1. A)A) 1, 2
    2. B)B) -1, 3
    3. C)C) 0, 3
    4. D)D) 0, -3
    Show answer

    Answer: C) 0, 3

    Hint: For real and equal roots, the discriminant (D) of a quadratic equation `ax^2 + bx + c = 0` must be equal to zero. Remember to check for the condition `a ≠ 0`.

  3. Q3.easy

    If `α` and `β` are the roots of the quadratic equation `x^2 - 5x + 3 = 0`, find the value of `α^2 + β^2`.
    1. A)A) 19
    2. B)B) 25
    3. C)C) 13
    4. D)D) 22
    Show answer

    Answer: A) 19

    Hint: Use the relationships between the roots and coefficients of a quadratic equation. Remember that `α^2 + β^2` can be expressed in terms of `(α+β)` and `αβ`.

  4. Q4.medium

    If the roots of the equation x^2 - 10x + k = 0 are p and q, and p^2 + q^2 = 52, find the value of k.
    1. A)12
    2. B)18
    3. C)24
    4. D)36
    Show answer

    Answer: 24

    Hint: Relate the sum and product of roots (p+q and pq) to the coefficients of the quadratic equation. Then, express p^2 + q^2 in terms of (p+q) and pq.

  5. Q5.medium

    If the roots of the quadratic equation px^2 + qx + r = 0 are reciprocal to each other, which of the following is true?
    1. A)p = q
    2. B)p = r
    3. C)q = r
    4. D)p + q + r = 0
    Show answer

    Answer: p = r

    Hint: If one root is α, what is the other root? Use Vieta's formulas to relate the product of roots to the coefficients.

  6. Q6.medium

    If α and β are the roots of the quadratic equation x^2 - 4x + 7 = 0, find the value of (α^2 + 1)(β^2 + 1).
    1. A)48
    2. B)50
    3. C)51
    4. D)52
    Show answer

    Answer: 52

    Hint: Expand the expression (α^2 + 1)(β^2 + 1) and relate it to the sum and product of roots. Alternatively, note that since α and β are roots, α^2 - 4α + 7 = 0 can simplify α^2.

  7. Q7.hard

    If α and β are the roots of the quadratic equation x² - 2x + 5 = 0, find the value of α⁴ + β⁴.
    1. A)-14
    2. B)22
    3. C)36
    4. D)50
    Show answer

    Answer: -14

    Hint: Use Vieta's formulas to find α+β and αβ. Then, express α⁴ + β⁴ in terms of these symmetric sums.

  8. Q8.hard

    Consider the quadratic equation x² + (k-2)x - k - 1 = 0. If one root is the negative reciprocal of the other, what is the value of k?
    1. A)0
    2. B)1
    3. C)-1
    4. D)2
    Show answer

    Answer: 0

    Hint: If one root is r, its negative reciprocal is -1/r. Use the relationship between the product of roots and the coefficients of the quadratic equation.

  9. Q9.hard

    The number of real solutions to the equation sqrt(x + 5) + sqrt(x) = 5 is:
    1. A)0
    2. B)1
    3. C)2
    4. D)3
    Show answer

    Answer: 1

    Hint: Isolate one of the square root terms before squaring to simplify the equation. Remember to check for extraneous solutions.

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.