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About Advanced Algebra — Class 10 IB

Solve quadratic equations, work with rational expressions, and explore polynomial functions. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 2). On this page you can practice 51 questions across three difficulty levels — 20 easy, 20 medium, and 11 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Advanced Algebra — solved examples for Class 10 IB

Example 1easy

Rohan was solving the quadratic equation (x - 3)(x + 2) = 6. He concluded that x - 3 = 6 or x + 2 = 6. Which statement best describes Rohan's error?
  1. A)The Zero Product Property only applies when one side of the equation is 0.
  2. B)He made an algebraic error in expanding the factors.
  3. C)He should have divided both sides by 6 first.
  4. D)He should have set x - 3 = 1 and x + 2 = 6, or x - 3 = 2 and x + 2 = 3.

Step-by-step solution

  1. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
  2. Rohan's equation is (x - 3)(x + 2) = 6, not (x - 3)(x + 2) = 0. Therefore, he cannot directly set each factor equal to 6.
  3. To solve this correctly, Rohan should first expand (x - 3)(x + 2) to get x² - x - 6, and then rearrange the equation to x² - x - 6 = 6, which becomes x² - x - 12 = 0. Only then can he factor or use other methods.

Answer: The Zero Product Property only applies when one side of the equation is 0.

Example 2medium

Solve the quadratic equation: 6x² - 19x + 10 = 0.
  1. A)A) x = 2/3, x = -5/2
  2. B)B) x = 2/3, x = 5/2
  3. C)C) x = 3/2, x = 5/2
  4. D)D) x = -2/3, x = -5/2

Step-by-step solution

  1. Multiply the coefficient of x² (6) by the constant term (10) to get 60. Find two numbers that multiply to 60 and add to -19. These numbers are -15 and -4.
  2. Rewrite the middle term: 6x² - 15x - 4x + 10 = 0.
  3. Factor by grouping: 3x(2x - 5) - 2(2x - 5) = 0, which simplifies to (3x - 2)(2x - 5) = 0.
  4. Set each factor to zero: 3x - 2 = 0 or 2x - 5 = 0. This gives x = 2/3 or x = 5/2.

Answer: B) x = 2/3, x = 5/2

Example 3hard

The quadratic equation (k² - 4)x² + (k + 2)x - 1 = 0 has real and distinct roots. What is the range of possible values for k?
  1. A)k < -2 or k > 2
  2. B)k < -2 or k > 6/5, k ≠ 2
  3. C)k > 6/5
  4. D)k < -2 or k > 6/5

Step-by-step solution

  1. For real and distinct roots, the discriminant Δ = b² - 4ac must be greater than 0. Here, a = k² - 4, b = k + 2, c = -1.
  2. Substitute the coefficients into the discriminant inequality: (k + 2)² - 4(k² - 4)(-1) > 0. This simplifies to k² + 4k + 4 + 4k² - 16 > 0, which further simplifies to 5k² + 4k - 12 > 0.
  3. Find the roots of 5k² + 4k - 12 = 0 using the quadratic formula: k = [-4 ± √(4² - 4(5)(-12))] / (2 × 5) = [-4 ± √(16 + 240)] / 10 = [-4 ± √256] / 10 = [-4 ± 16] / 10. The roots are k₁ = (-4 - 16)/10 = -2 and k₂ = (-4 + 16)/10 = 12/10 = 6/5. Since 5k² + 4k - 12 > 0, the solution is k < -2 or k > 6/5.
  4. Also, for the equation to be quadratic, the coefficient of x² cannot be zero: k² - 4 ≠ 0, meaning (k - 2)(k + 2) ≠ 0. So, k ≠ 2 and k ≠ -2. The condition k < -2 or k > 6/5 already excludes k = -2. Thus, the final range is k < -2 or k > 6/5.

Answer: k < -2 or k > 6/5

Practice questions on Advanced Algebra

  1. Q1.easy

    Consider a quadratic equation ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. Which of the following statements about its roots is FALSE?
    1. A)If b² - 4ac < 0, the equation has two distinct real roots.
    2. B)If b² - 4ac = 0, the equation has two equal real roots.
    3. C)If b² - 4ac > 0, the equation has two distinct real roots.
    4. D)If b² - 4ac < 0, the equation has no real roots.
    Show answer

    Answer: If b² - 4ac < 0, the equation has two distinct real roots.

    Hint: Recall how the discriminant (b² - 4ac) relates to the nature of the roots. One of the options describes a condition incorrectly.

  2. Q2.easy

    If α and β are the roots of the quadratic equation 2x² - 5x + 3 = 0, what is the value of α + β - αβ?
    1. A)5/2
    2. B)3/2
    3. C)4
    4. D)1
    Show answer

    Answer: 1

    Hint: Remember the formulas for the sum and product of roots in terms of the coefficients of a quadratic equation ax² + bx + c = 0.

  3. Q3.easy

    To solve the equation x² + 6x - 7 = 0 by completing the square, the first step is to move the constant term to the right side: x² + 6x = 7. What should be the NEXT correct step?
    1. A)Add (6/2)² to both sides.
    2. B)Subtract (6/2)² from both sides.
    3. C)Factor x² + 6x as x(x + 6).
    4. D)Take the square root of both sides.
    Show answer

    Answer: Add (6/2)² to both sides.

    Hint: To complete the square for x² + bx, you need to add (b/2)² to both sides of the equation.

  4. Q4.medium

    Find the values of x for the equation 3x² - 7x + 1 = 0.
    1. A)A) (7 ± √61) / 6
    2. B)B) (-7 ± √37) / 6
    3. C)C) (7 ± √37) / 3
    4. D)D) (7 ± √37) / 6
    Show answer

    Answer: D) (7 ± √37) / 6

    Hint: Recall the quadratic formula x = (-b ± √(b² - 4ac)) / (2a) for an equation ax² + bx + c = 0.

  5. Q5.medium

    Find the values of k for which the quadratic equation x² + (k-2)x + (k+1) = 0 has real and equal roots.
    1. A)A) k = 4, k = 8
    2. B)B) k = 0, k = 8
    3. C)C) k = 2, k = -1
    4. D)D) k = 0, k = -8
    Show answer

    Answer: B) k = 0, k = 8

    Hint: For real and equal roots, the discriminant (b² - 4ac) must be equal to zero.

  6. Q6.medium

    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)A) 8 km/h
    2. B)B) 4 km/h
    3. C)C) 6 km/h
    4. D)D) 3 km/h
    Show answer

    Answer: C) 6 km/h

    Hint: Define the speed of the stream as a variable and use the formula time = distance / speed for both upstream and downstream journeys. Remember that upstream speed subtracts stream speed, and downstream speed adds it.

  7. Q7.hard

    If α and β are the roots of the quadratic equation 3x² - 4x + 1 = 0, find the value of (α³/β + β³/α).
    1. A)16/9
    2. B)82/27
    3. C)100/81
    4. D)4/3
    Show answer

    Answer: 82/27

    Hint: Start by finding the sum (α + β) and product (αβ) of the roots. Then, express the given expression using these fundamental symmetric polynomials and relevant algebraic identities.

  8. Q8.hard

    Solve the equation √(x² - 3x + 3) + √(x² - 3x + 6) = 3.
    1. A)x = 0, x = 3
    2. B)x = -1, x = -2
    3. C)x = 1, x = 2
    4. D)No real solutions
    Show answer

    Answer: x = 1, x = 2

    Hint: Look for a common expression within the square roots and use a substitution to simplify the equation. Be sure to check your solutions in the original equation to avoid extraneous roots.

  9. Q9.hard

    Consider the equation: (x² - x - 2) / (x² + x - 6) = (x² + 2x + 1) / (x² + 5x + 4). Find the sum of all valid solutions for x.
    1. A)0
    2. B)-1
    3. C)1
    4. D)2
    Show answer

    Answer: 0

    Hint: Factorize all quadratic expressions in the numerators and denominators first. This will help simplify the rational expressions and identify any domain restrictions before solving.

These are 9 of the 51 questions available for Advanced Algebra. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.