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About Quadratic Expressions — Class 9 IB

Expand, factorize, and simplify quadratic expressions; complete the square. This topic is part of the IB Class 9 mathematics syllabus (chapter: Unit 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.

Quadratic Expressions — solved examples for Class 9 IB

Example 1easy

Which of the following statements correctly describes a quadratic expression?
  1. A)An expression where the highest power of the variable is 1.
  2. B)An expression with exactly two terms.
  3. C)An expression where the highest power of the variable is 2, and the coefficient of the squared term is not zero.
  4. D)An expression that always has three terms.

Step-by-step solution

  1. A quadratic expression is defined by the highest power of its variable being 2.
  2. The coefficient of the x² term (a) must not be zero; otherwise, it would not be quadratic.
  3. Options A describes a linear expression. Options B and D are not universally true for all quadratic expressions (e.g., 2x² is quadratic but has one term, x² + 5x is quadratic but has two terms).

Answer: An expression where the highest power of the variable is 2, and the coefficient of the squared term is not zero.

Example 2medium

Expand and simplify the expression (x + 7)(x - 4).
  1. A)x² + 3x + 28
  2. B)x² + 3x - 28
  3. C)x² - 3x - 28
  4. D)x² - 3x + 28

Step-by-step solution

  1. Apply the distributive property (FOIL method): First, Outer, Inner, Last.
  2. x × x = x²
  3. x × (-4) = -4x
  4. 7 × x = 7x
  5. 7 × (-4) = -28
  6. Combine the terms: x² - 4x + 7x - 28 = x² + 3x - 28

Answer: x² + 3x - 28

Example 3hard

Expand and simplify the expression (2x - y + 3)² - (x + 2y - 1)².
  1. A)3x² + 3y² + 14x + 2y - 8xy + 8
  2. B)x² - 5y² + 14x - 4y - 8xy + 10
  3. C)3x² - 3y² + 14x - 2y - 8xy + 8
  4. D)x² - y² + 10x - 6y - 6xy + 8

Step-by-step solution

  1. Let A = (2x - y + 3) and B = (x + 2y - 1). The expression is A² - B² = (A - B)(A + B).
  2. Calculate (A - B): (2x - y + 3) - (x + 2y - 1) = 2x - y + 3 - x - 2y + 1 = x - 3y + 4.
  3. Calculate (A + B): (2x - y + 3) + (x + 2y - 1) = 2x - y + 3 + x + 2y - 1 = 3x + y + 2.
  4. Multiply the results: (x - 3y + 4)(3x + y + 2) = 3x² + xy + 2x - 9xy - 3y² - 6y + 12x + 4y + 8 = 3x² - 3y² + 14x - 2y - 8xy + 8.

Answer: 3x² - 3y² + 14x - 2y - 8xy + 8

Practice questions on Quadratic Expressions

  1. Q1.easy

    Ravi was asked to expand (x + 4)(x + 7). He wrote his solution as x² + 28. What mistake did Ravi make?
    1. A)He did not multiply the x terms correctly.
    2. B)He forgot to add the middle terms (4x and 7x).
    3. C)He multiplied 4 and 7 incorrectly.
    4. D)He should have subtracted the constants instead of multiplying them.
    Show answer

    Answer: He forgot to add the middle terms (4x and 7x).

    Hint: Remember to apply the distributive property (often called FOIL) completely, ensuring every term in the first bracket multiplies every term in the second.

  2. Q2.easy

    When expanding the expression (3x - 5)², a common error is to write it as 9x² - 25. Which statement best explains why this is incorrect?
    1. A)The constant term should be negative.
    2. B)The coefficient of x² should be 3, not 9.
    3. C)The middle term (-2ab) is missing.
    4. D)The terms should be multiplied, not squared.
    Show answer

    Answer: The middle term (-2ab) is missing.

    Hint: Recall the algebraic identity for a perfect square: (a - b)² = a² - 2ab + b².

  3. Q3.easy

    To factorize the expression 6x² + 9x, what is the most appropriate first step?
    1. A)Use the difference of squares formula.
    2. B)Look for a common factor among the terms.
    3. C)Apply the perfect square trinomial formula.
    4. D)Try to split the middle term.
    Show answer

    Answer: Look for a common factor among the terms.

    Hint: Always begin factorization by checking if there's a common factor that can be taken out from all terms.

  4. Q4.medium

    Expand and simplify the expression (3x - 5)².
    1. A)9x² - 25
    2. B)9x² + 25
    3. C)9x² - 15x + 25
    4. D)9x² - 30x + 25
    Show answer

    Answer: 9x² - 30x + 25

    Hint: Recall the algebraic identity for squaring a binomial: (a - b)² = a² - 2ab + b².

  5. Q5.medium

    Factorize the expression 100a² - 121b².
    1. A)(10a - 11b)(10a + 11b)
    2. B)(10a - 11b)²
    3. C)(10a + 11b)²
    4. D)(50a - 60.5b)(2a + 2b)
    Show answer

    Answer: (10a - 11b)(10a + 11b)

    Hint: This expression is in the form of a difference of two squares. Remember the identity a² - b² = (a - b)(a + b).

  6. Q6.medium

    Factorize the quadratic expression x² - 9x + 18.
    1. A)(x + 3)(x + 6)
    2. B)(x - 2)(x - 9)
    3. C)(x - 3)(x - 6)
    4. D)(x + 2)(x - 9)
    Show answer

    Answer: (x - 3)(x - 6)

    Hint: Look for two numbers that multiply to give the constant term (18) and add up to the coefficient of the x term (-9).

  7. Q7.hard

    Factorize completely the expression 6x³ + 7x² - 5x.
    1. A)x(6x - 5)(x + 1)
    2. B)x(2x - 1)(3x + 5)
    3. C)x(3x - 1)(2x + 5)
    4. D)x(2x + 1)(3x - 5)
    Show answer

    Answer: x(2x - 1)(3x + 5)

    Hint: Always look for a common factor first before attempting to factorize the quadratic trinomial. Then apply grouping or splitting the middle term.

  8. Q8.hard

    Simplify the algebraic fraction: (2x² + 5x - 3) / (4x² - 1).
    1. A)(2x - 1) / (x + 3)
    2. B)(x - 3) / (2x + 1)
    3. C)(x + 3) / (2x - 1)
    4. D)(x + 3) / (2x + 1)
    Show answer

    Answer: (x + 3) / (2x + 1)

    Hint: Factorize both the numerator and the denominator completely. Look for common factors to cancel out.

  9. Q9.hard

    If the expression 3x² - 12x + k can be written in the form a(x - b)² + 5, find the value of k.
    1. A)17
    2. B)14
    3. C)11
    4. D)20
    Show answer

    Answer: 17

    Hint: Complete the square for the given expression and then compare it to the target form to find the value of k.

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