Question Simulator

IGCSE Additional Mathematics (0606)

Quadratic functions

8 questions · about 2% of the marks we indexed · examined in 7 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 22 · Q1 (a)2 marksQuadratic graphs and completing the square
  • 2026 May/June · Paper 22 · Q23 marksDiscriminant and nature of roots

2025

  • 2025 Oct/Nov · Paper 12 · Q4 (a, b)4 marksQuadratic graphs and completing the square

2024

  • 2024 May/June · Paper 22 · Q3 (a)4 marksDiscriminant and nature of roots

2023

  • 2023 Oct/Nov · Paper 22 · Q25 marksDiscriminant and nature of roots
  • 2023 May/June · Paper 22 · Q1 (a, b(i), b(ii))7 marksQuadratic graphs and completing the square · Quadratic inequalities

2022

  • 2022 Oct/Nov · Paper 12 · Q2 (a)4 marksQuadratic graphs and completing the square
  • 2022 May/June · Paper 22 · Q35 marksDiscriminant and nature of roots

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Discriminant and nature of roots · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Find the values of k for which the equation x² + kx + 9 = 0 has equal roots. [2]
Show answer
  1. (a) k = 6 or k = −6
    k² − 36 = 0 M1; k = ±6 A1
Q2Discriminant and nature of roots · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Show that the equation 2x² − 3x + 4 = 0 has no real roots. [2]
Show answer
  1. (a) Shown: discriminant = −23 < 0
    b² − 4ac = 9 − 32 M1; = −23 < 0 A1
Q3Discriminant and nature of roots · Core · Paper 2 style · calculator · 5 marks

A ball is kicked upwards in a school playground in Quetta. Its height, h metres, after t seconds is h = 18t − 5t².

  1. (a) By using the discriminant, show that the ball does not reach a height of 20 m. [3]
  2. (b) Find the greatest height reached by the ball. [2]
Show answer
  1. (a) Shown: discriminant −76 < 0
    5t² − 18t + 20 = 0 M1; 324 − 400 = −76 M1; < 0 so no real t A1
  2. (b) 16.2 m
    18t − 5t² = −5(t − 1.8)² + 16.2 M1; 16.2 m A1
Q4Discriminant and nature of roots · Core · Paper 1 style · no calculator · 3 marks
  1. (a) State, with a reason, the nature of the roots of 4x² − 12x + 9 = 0. [1]
  2. (b) State, with a reason, the nature of the roots of 3x² + 5x − 1 = 0. [1]
  3. (c) State, with a reason, the nature of the roots of x² + x + 2 = 0. [1]
Show answer
  1. (a) Equal (repeated) roots, since b² − 4ac = 0
    B1
  2. (b) Two distinct real roots, since b² − 4ac = 37 > 0
    B1
  3. (c) No real roots, since b² − 4ac = −7 < 0
    B1
Q5Discriminant and nature of roots · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Find the values of m for which the line y = mx + 1 is a tangent to the curve y = x² + 3x + 5. [4]
Show answer
  1. (a) m = −1 or m = 7
    x² + (3 − m)x + 4 = 0 M1; (3 − m)² − 16 = 0 M1 A1; 3 − m = ±4 → m = −1, 7 A1
Q6Discriminant and nature of roots · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Find the set of values of c for which the line y = 2x + c does not meet the curve y = x² − 4x + 7. [4]
Show answer
  1. (a) c < −2
    x² − 6x + 7 − c = 0 M1; 36 − 4(7 − c) < 0 M1 A1; c < −2 A1
Q7Discriminant and nature of roots · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) Do not use a calculator in this question.Find the set of values of k for which the equation x² − (k − 2)x + 2k − 3 = 0 has two distinct real roots. Give the boundary values in exact form. [5]
Show answer
  1. (a) k < 6 − 2√5 or k > 6 + 2√5
    (k − 2)² − 4(2k − 3) > 0 M1; k² − 12k + 16 > 0 A1; critical values k = (12 ± √80)/2 = 6 ± 2√5 M1 A1; outside the roots A1
Q8Discriminant and nature of roots · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Show that the equation x² + kx + k − 2 = 0 has two distinct real roots for all real values of k. [3]
Show answer
  1. (a) Shown: discriminant = (k − 2)² + 4 > 0
    k² − 4(k − 2) = k² − 4k + 8 M1; = (k − 2)² + 4 M1; > 0 for all k A1
Q9Discriminant and nature of roots · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) The curve y = 2x² + px + 8 touches the x-axis. Find the possible values of p and, for each value, the coordinates of the point where the curve touches the x-axis. [5]
Show answer
  1. (a) p = 8, point (−2, 0); p = −8, point (2, 0)
    p² − 64 = 0 M1; p = ±8 A1; repeated root x = −p/4 M1; p = 8: (−2, 0) A1; p = −8: (2, 0) A1
Q10Discriminant and nature of roots · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Find the values of p for which (p − 1)x² + 4x + (p + 2) = 0 has equal roots, and find the root in each case. [6]
Show answer
  1. (a) p = 2, root x = −2; p = −3, root x = ½
    16 − 4(p − 1)(p + 2) = 0 M1; p² + p − 6 = 0 A1; p = 2 or −3 A1; p = 2: x² + 4x + 4 = 0 → x = −2 M1 A1; p = −3: −4x² + 4x − 1 = 0 → x = ½ A1
Q11Discriminant and nature of roots · Standard · Paper 2 style · calculator · 4 marks
  1. (a) Show that the line y = 3x − 5 is a tangent to the curve y = x² − x − 1, and find the coordinates of the point of contact. [4]
Show answer
  1. (a) Shown; point of contact (2, 1)
    x² − 4x + 4 = 0 M1; discriminant 16 − 16 = 0 so tangent A1; (x − 2)² = 0 → x = 2 M1; y = 1 A1
Q12Discriminant and nature of roots · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) Show that the equation kx² + (k + 3)x + 1 = 0, where k is a non-zero constant, has two distinct real roots for every value of k. [4]
Show answer
  1. (a) Shown: discriminant = (k + 1)² + 8 > 0
    b² − 4ac = (k + 3)² − 4k M1; = k² + 2k + 9 A1; = (k + 1)² + 8 M1; always positive, so two distinct real roots A1
Q13Discriminant and nature of roots · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Do not use a calculator in this question.Find the set of values of k for which the line y = kx − 3 meets the curve y = x² + 2x + k. [5]
Show answer
  1. (a) k ≤ 4 − 2√6 or k ≥ 4 + 2√6
    x² + (2 − k)x + k + 3 = 0 M1; (2 − k)² − 4(k + 3) ≥ 0 M1; k² − 8k − 8 ≥ 0 A1; k = 4 ± 2√6 M1; k ≤ 4 − 2√6 or k ≥ 4 + 2√6 A1
Q14Discriminant and nature of roots · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) Find the set of values of k for which x² + 2kx + 3k + 4 > 0 for all real values of x. [4]
Show answer
  1. (a) −1 < k < 4
    Need discriminant < 0: 4k² − 4(3k + 4) < 0 M1; k² − 3k − 4 < 0 A1; (k − 4)(k + 1) < 0 M1; −1 < k < 4 A1
Q15Discriminant and nature of roots · Challenge · Paper 2 style · calculator · 6 marks

A farmer in Punjab uses 100 m of fencing to make the four sides of a rectangular pen. One side of the pen is x m long.

  1. (a) Show that the area of the pen is (50x − x²) m². [1]
  2. (b) Use the discriminant to show that the area cannot be 700 m². [3]
  3. (c) Find the greatest possible area. [2]
Show answer
  1. (a) Shown
    Other side = 50 − x B1
  2. (b) Shown: discriminant −300 < 0
    x² − 50x + 700 = 0 M1; 2500 − 2800 = −300 M1; < 0, no real x A1
  3. (c) 625 m²
    50x − x² = 625 − (x − 25)² M1; 625 A1
Q16Quadratic graphs and completing the square · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Express x² − 6x + 14 in the form (x − a)² + b. [2]
  2. (b) Hence state the minimum value of x² − 6x + 14 and the value of x at which it occurs. [2]
Show answer
  1. (a) (x − 3)² + 5
    a = 3 B1; b = 5 B1
  2. (b) minimum 5 when x = 3
    B1 B1
Q17Quadratic graphs and completing the square · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Write 2x² + 8x − 3 in the form a(x + b)² + c. [3]
  2. (b) State the coordinates of the vertex of y = 2x² + 8x − 3. [1]
Show answer
  1. (a) 2(x + 2)² − 11
    a = 2 B1; b = 2 B1; c = −11 B1
  2. (b) (−2, −11)
    B1
Q18Quadratic graphs and completing the square · Core · Paper 1 style · no calculator · 5 marks
  1. (a) A quadratic curve crosses the x-axis at x = −1 and x = 4 and crosses the y-axis at (0, −8). Find its equation in the form y = ax² + bx + c. [3]
  2. (b) Find the coordinates of the vertex of the curve. [2]
Show answer
  1. (a) y = 2x² − 6x − 8
    y = a(x + 1)(x − 4) M1; −8 = −4a → a = 2 A1; expand A1
  2. (b) (1.5, −12.5)
    x = (−1 + 4)/2 = 1.5 M1; y = 2(2.25) − 9 − 8 = −12.5 A1
Q19Quadratic graphs and completing the square · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Express 5 + 4x − x² in the form p − (x − q)². [2]
  2. (b) Hence state the maximum value of f(x) = 5 + 4x − x² and the range of f for all real x. [2]
  3. (c) Hence solve 5 + 4x − x² = 0. [2]
Show answer
  1. (a) 9 − (x − 2)²
    q = 2 B1; p = 9 B1
  2. (b) maximum 9 (at x = 2); range f(x) ≤ 9
    B1 B1
  3. (c) x = 5, x = −1
    (x − 2)² = 9 M1; x = 2 ± 3 A1
Q20Quadratic graphs and completing the square · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Express 3x² − 12x + 7 in the form a(x − b)² + c. [3]
  2. (b) Do not use a calculator in this part.Hence solve 3x² − 12x + 7 = 0, giving your answers in the form p ± q√15, where p and q are rational. [3]
Show answer
  1. (a) 3(x − 2)² − 5
    a = 3, b = 2, c = −5 B1 B1 B1
  2. (b) x = 2 ± (1/3)√15
    (x − 2)² = 5/3 M1; x − 2 = ±√(5/3) M1; √(5/3) = √15/3 so x = 2 ± √15/3 A1
Q21Quadratic graphs and completing the square · Standard · Paper 1 style · no calculator · 6 marks
  1. (a) Express x² − 4x − 5 in the form (x − a)² − b. [2]
  2. (b) Describe the graph of y = x² − 4x − 5 by stating the coordinates of its vertex and of the points where it meets the axes. [4]
Show answer
  1. (a) (x − 2)² − 9
    B1 B1
  2. (b) vertex (2, −9); x-axis (−1, 0) and (5, 0); y-axis (0, −5)
    vertex B1; (x − 2)² = 9 gives x = −1, 5 M1 A1; (0, −5) B1
Q22Quadratic graphs and completing the square · Standard · Paper 2 style · calculator · 6 marks

A company in Sialkot makes footballs. Its weekly profit, P thousand rupees, from selling x hundred footballs is P = −2x² + 48x − 160.

  1. (a) Write P in the form −2(x − a)² + b. [2]
  2. (b) State the maximum weekly profit and the number of footballs sold to achieve it. [2]
  3. (c) Find the values of x for which the company breaks even (P = 0). [2]
Show answer
  1. (a) P = −2(x − 12)² + 128
    a = 12 B1; b = 128 B1
  2. (b) 128 thousand rupees when 1200 footballs are sold
    B1 B1
  3. (c) x = 4 and x = 20 (400 and 2000 footballs)
    (x − 12)² = 64 M1; x = 4, 20 A1
Q23Quadratic graphs and completing the square · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) A quadratic curve has its vertex at (3, −2) and passes through the point (5, 6). Find its equation in the form y = ax² + bx + c. [4]
Show answer
  1. (a) y = 2x² − 12x + 16
    y = a(x − 3)² − 2 M1; 6 = 4a − 2 → a = 2 A1; expand 2(x² − 6x + 9) − 2 M1; y = 2x² − 12x + 16 A1
Q24Quadratic graphs and completing the square · Standard · Paper 2 style · calculator · 5 marks

A cricket ball is hit upwards. Its height, h metres, t seconds after being hit is h = 1.5 + 14.7t − 4.9t².

  1. (a) Write h in the form p − 4.9(t − q)². [3]
  2. (b) Hence state the greatest height of the ball, correct to 3 significant figures, and the time at which it occurs. [2]
Show answer
  1. (a) h = 12.525 − 4.9(t − 1.5)²
    factor −4.9 from the t terms: −4.9(t² − 3t) M1; q = 1.5 A1; p = 1.5 + 4.9 × 2.25 = 12.525 A1
  2. (b) 12.5 m at t = 1.5 s
    B1 B1
Q25Quadratic graphs and completing the square · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Express 4x² − 12x + 1 in the form (2x − a)² − b. [2]
  2. (b) Hence find the range of f(x) = 4x² − 12x + 1 for the domain 0 ≤ x ≤ 4. [3]
Show answer
  1. (a) (2x − 3)² − 8
    a = 3 B1; b = 8 B1
  2. (b) −8 ≤ f(x) ≤ 17
    minimum −8 at x = 1.5 (inside the domain) B1; f(0) = 1, f(4) = 17 M1; range −8 ≤ f(x) ≤ 17 A1
Q26Quadratic graphs and completing the square · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Find the values of a, b and c such that 7 − 6x − 3x² ≡ a − b(x + c)². [3]
  2. (b) Hence state the coordinates of the stationary point of y = 7 − 6x − 3x², and whether it is a maximum or a minimum. [2]
Show answer
  1. (a) a = 10, b = 3, c = 1
    b = 3 B1; c = 1 B1; a = 10 B1
  2. (b) (−1, 10), a maximum
    B1 B1
Q27Quadratic graphs and completing the square · Challenge · Paper 1 style · no calculator · 5 marks

f(x) = x² + 2kx + 5, where k is a constant.

  1. (a) Express f(x) in the form (x + p)² + q, giving p and q in terms of k. [2]
  2. (b) Find the values of k for which the minimum value of f(x) is 1. [2]
  3. (c) When k = 3, state whether the curve y = f(x) meets the x-axis, giving a reason. [1]
Show answer
  1. (a) (x + k)² + 5 − k²
    p = k B1; q = 5 − k² B1
  2. (b) k = 2 or k = −2
    5 − k² = 1 M1; k = ±2 A1
  3. (c) Yes: the minimum value is 5 − 9 = −4 < 0, so the curve crosses the x-axis twice
    B1
Q28Quadratic inequalities · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the inequality x² − 5x − 14 < 0. [3]
Show answer
  1. (a) −2 < x < 7
    (x − 7)(x + 2) M1; critical values −2, 7 A1; −2 < x < 7 A1
Q29Quadratic inequalities · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Solve the inequality 2x² + x ≥ 15. [3]
Show answer
  1. (a) x ≤ −3 or x ≥ 5/2
    2x² + x − 15 ≥ 0 → (2x − 5)(x + 3) ≥ 0 M1; critical values −3, 5/2 A1; x ≤ −3 or x ≥ 5/2 A1
Q30Quadratic inequalities · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Find the set of values of x that satisfy both x² − 3x − 10 ≤ 0 and 3x² > 2x + 8. [5]
Show answer
  1. (a) −2 ≤ x < −4/3 or 2 < x ≤ 5
    (x − 5)(x + 2) ≤ 0 → −2 ≤ x ≤ 5 M1 A1; (3x + 4)(x − 2) > 0 → x < −4/3 or x > 2 M1 A1; combine: −2 ≤ x < −4/3 or 2 < x ≤ 5 A1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Discriminant and nature of roots 4 questions · 4/9 sessions
Quadratic graphs and completing the square 4 questions · 4/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Discriminant and nature of roots 174/9
Quadratic graphs and completing the square 144/9
Quadratic inequalities Rare31/9

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