Loading...

About Quadratic Equations — Class 10 ICSE

Solve quadratic equations by factorization, completing the square, and the quadratic formula. This topic is part of the ICSE Class 10 mathematics syllabus (chapter: Chapter 2). 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.

What you'll learn in Quadratic Equations

  • Introduction to Quadratic Equations
  • Solving Quadratic Equations by Factorization
  • Solving Quadratic Equations using the Quadratic Formula
  • Completing the Square and Nature of Roots
  • Summary, Connections, and Exam Preparation

Interactive lesson · about 15 minutes · checkpoint question after every unit

Quadratic Equations — solved examples for Class 10 ICSE

Example 1easy

A student states that any equation containing an x² term is a quadratic equation. Which of the following statements correctly identifies the flaw in this reasoning for the standard form ax² + bx + c = 0?
  1. A)It must also have an 'x' term.
  2. B)The coefficient of x² (i.e., 'a') cannot be zero.
  3. C)It must always be equal to zero.
  4. D)It must have only two terms.

Step-by-step solution

  1. The standard form of a quadratic equation is ax² + bx + c = 0. A crucial condition for it to be quadratic is that the coefficient of the x² term, 'a', must not be zero.
  2. If a = 0, the x² term vanishes, and the equation reduces to bx + c = 0, which is a linear equation, not a quadratic one.

Answer: The coefficient of x² (i.e., 'a') cannot be zero.

Example 2medium

Which of the following equations, when simplified, is NOT a quadratic equation?
  1. A)A) (x + 1)² = 2(x - 3)
  2. B)B) x(x + 5) = x² + 3x - 1
  3. C)C) (2x - 1)(x + 3) = x(x - 2)
  4. D)D) (x - 2)³ = x³ - 4x

Step-by-step solution

  1. Simplify option A: (x + 1)² = 2(x - 3) ⇒ x² + 2x + 1 = 2x - 6 ⇒ x² + 7 = 0. This is a quadratic equation (coefficient of x² is 1).
  2. Simplify option B: x(x + 5) = x² + 3x - 1 ⇒ x² + 5x = x² + 3x - 1 ⇒ 2x + 1 = 0. This is a linear equation, not a quadratic equation, as the x² term cancels out.
  3. Simplify option C: (2x - 1)(x + 3) = x(x - 2) ⇒ 2x² + 6x - x - 3 = x² - 2x ⇒ 2x² + 5x - 3 = x² - 2x ⇒ x² + 7x - 3 = 0. This is a quadratic equation.
  4. Simplify option D: (x - 2)³ = x³ - 4x ⇒ x³ - 6x² + 12x - 8 = x³ - 4x ⇒ -6x² + 16x - 8 = 0. This is a quadratic equation (coefficient of x² is -6).

Answer: B) x(x + 5) = x² + 3x - 1

Example 3hard

Solve the quadratic equation 4x² - 4a²x + (a⁴ - b⁴) = 0 for x.
  1. A)(a²+b²)/4, (a²-b²)/4
  2. B)(a²+b²)/2, (a²-b²)/2
  3. C)(a²-b²)/2, (a⁴-b⁴)/4
  4. D)(a²+b²)/2, (a-b)/2

Step-by-step solution

  1. The given equation is 4x² - 4a²x + (a⁴ - b⁴) = 0. This is in the form Ax² + Bx + C = 0, where A=4, B=-4a², C=a⁴-b⁴.
  2. Using the quadratic formula x = [-B ± √(B² - 4AC)] / 2A:
    x=[−(−4a2)±√((−4a2)2−4(4)(a4−b4))]/(2×4)x = [ -(-4a²) ± √((-4a²)² - 4(4)(a⁴ - b⁴)) ] / (2 × 4)
  3. Simplify the expression under the square root (discriminant):
    D=16a4−16(a4−b4)=16a4−16a4+16b4=16b4D = 16a⁴ - 16(a⁴ - b⁴) = 16a⁴ - 16a⁴ + 16b⁴ = 16b⁴
  4. Substitute D back into the formula:
    x=[4a2±√(16b4)]/8=[4a2±4b2]/8=(a2±b2)/2x = [ 4a² ± √(16b⁴) ] / 8 = [ 4a² ± 4b² ] / 8 = (a² ± b²) / 2

Answer: (a²+b²)/2, (a²-b²)/2

Practice questions on Quadratic Equations

  1. Q1.easy

    Rohan claims that x = 2 is a root of the equation 3x² - 7x + 2 = 0. To verify this, he substituted x = 2 into the equation and calculated 3(2)² - 7(2) + 2 = 12 - 14 + 2 = 0. He concluded it is a root. Is his reasoning correct?
    1. A)Yes, because substituting the value results in 0, satisfying the equation.
    2. B)No, he should have factored the equation first to find the roots.
    3. C)No, he should have used the quadratic formula to find the roots.
    4. D)Yes, but only if the discriminant of the equation is positive.
    Show answer

    Answer: Yes, because substituting the value results in 0, satisfying the equation.

    Hint: Think about the definition of a 'root' or 'solution' of an equation. What does it mean for a value to be a root?

  2. Q2.easy

    To solve the quadratic equation (x - 3)(x + 5) = 0, a student correctly sets each factor to zero, i.e., x - 3 = 0 or x + 5 = 0. This method relies on which fundamental algebraic property?
    1. A)Distributive Property
    2. B)Commutative Property
    3. C)Zero Product Property
    4. D)Associative Property
    Show answer

    Answer: Zero Product Property

    Hint: Consider what must be true if the product of two or more numbers (or expressions) is equal to zero.

  3. Q3.easy

    For a quadratic equation ax² + bx + c = 0 (where a, b, c are real numbers and a ≠ 0), if the discriminant (D = b² - 4ac) is equal to zero, what can be concluded about its roots?
    1. A)The equation has two distinct real roots.
    2. B)The equation has two equal real roots.
    3. C)The equation has no real roots.
    4. D)The equation has one real root and one imaginary root.
    Show answer

    Answer: The equation has two equal real roots.

    Hint: Recall the three main cases for the discriminant's value (positive, zero, or negative) and what each implies about the nature of the roots.

  4. Q4.medium

    Solve the quadratic equation 6x² - 7x - 3 = 0 by factorization.
    1. A)A) x = 3/2, x = -1/3
    2. B)B) x = -3/2, x = 1/3
    3. C)C) x = 1/2, x = -3/3
    4. D)D) x = 3, x = -1
    Show answer

    Answer: A) x = 3/2, x = -1/3

    Hint: Look for two numbers whose product is (6 × -3) = -18 and whose sum is -7 to split the middle term.

  5. Q5.medium

    Using the quadratic formula, find the roots of the equation 4x² - 12x + 3 = 0.
    1. A)A) (3 ± √3) / 2
    2. B)B) (3 ± √6) / 2
    3. C)C) (6 ± √3) / 4
    4. D)D) (6 ± √6) / 2
    Show answer

    Answer: B) (3 ± √6) / 2

    Hint: Remember the quadratic formula x = [-b ± √(b² - 4ac)] / 2a and simplify the square root carefully.

  6. Q6.medium

    Find the value of 'k' for which the quadratic equation kx(x - 2) + 6 = 0 has two equal real roots.
    1. A)A) k = 3
    2. B)B) k = 0
    3. C)C) k = 6
    4. D)D) k = -6
    Show answer

    Answer: C) k = 6

    Hint: First, rewrite the equation in the standard form ax² + bx + c = 0. Then, for equal real roots, the discriminant (D) must be zero.

  7. Q7.hard

    For what values of 'p' will the quadratic equation (p² - 1)x² + 2(p-1)x + 1 = 0 have real roots?
    1. A)p < 1
    2. B)p ≤ 1, p ≠ -1
    3. C)p < 1, p ≠ -1
    4. D)p ≥ 1
    Show answer

    Answer: p < 1, p ≠ -1

    Hint: For real roots, the discriminant must be non-negative. Also, remember the condition for an equation to be truly quadratic.

  8. Q8.hard

    If α and β are the roots of the quadratic equation x² - 3x + 1 = 0, then find the value of (α/β)² + (β/α)².
    1. A)45
    2. B)47
    3. C)49
    4. D)51
    Show answer

    Answer: 47

    Hint: First, find the sum (α+β) and product (αβ) of the roots. Then, express the given expression in terms of (α+β) and (αβ) using algebraic identities.

  9. Q9.hard

    Two pipes running together can fill a tank in 11 1/9 minutes. If one pipe takes 5 minutes more than the other to fill the tank separately, find the time taken by the pipe with the lesser filling rate to fill the tank separately.
    1. A)20 minutes
    2. B)25 minutes
    3. C)15 minutes
    4. D)30 minutes
    Show answer

    Answer: 25 minutes

    Hint: Let 'x' be the time taken by the faster pipe. Form an equation based on their combined work rate. Remember, a lesser filling rate means it takes more time.

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.