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Quadratic equations: OCR GCSE Maths questions

Every OCR GCSE Maths exam question on A18 Quadratic equations, 2017–2025, with its mark scheme and examiner’s report.

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921 questions

  1. Q9

    June 2025 · Paper 6

    6 marksA17 Linear equationsA18 Quadratic equations
    The whole paper ↗
    June 2025 Paper 6, question 9: (a) Solve. 8 x 11 6 x 3 - = + (a) x = (b) Solve by factorising. 2 + x 15 x 16 0 - = (b) x = or x =

  2. Q18

    November 2024 · Paper 5

    6 marksA18 Quadratic equationsA4 Simplifying, expanding and factorising
    The whole paper ↗
    November 2024 Paper 5, question 18: Solve. 2 x 5 - 4 x = x 4 - You must show your working. x = or x =

  3. Q20

    November 2024 · Paper 6

    6 marksA4 Simplifying, expanding and factorisingA18 Quadratic equations
    The whole paper ↗
    November 2024 Paper 6, question 20: ( 4 x a )( 4 x a x )( 2 ) 16 x bx 50 + - + = + - Find the two possible pairs of values for a and b. You must show your working. Pair 1: a = and b = Pair 2: a =

  4. Q15

    June 2024 · Paper 4

    10 marksA18 Quadratic equationsA11 Quadratic graphs and turning points
    The whole paper ↗
    June 2024 Paper 4, question 15: (a) Solve by factorisation. 2 + 3 x 10 x 8 0 - = (a) x = or x = 2 + 2 (b) Write x 8 x 11 in the form ( x a ) b . + + - (b) 2 (c) (i) Write down the coordinates

  5. Q8

    June 2024 · Paper 6

    6 marksA12 Recognising and sketching graphsA18 Quadratic equations
    The whole paper ↗
    June 2024 Paper 6, question 8: (a) Complete this table for y x 3 x . = - x 0 1 2 3 4 2 1 - - 0 16 y 20 2 4 - - - 3 2 (b) Draw the graph of y x 3 x for values of x from 2 - to 4. = - y 20 15 1

  6. Q18

    November 2023 · Paper 4

    4 marksA18 Quadratic equations
    The whole paper ↗
    November 2023 Paper 4, question 18: Solve this quadratic equation by factorisation. 2 - 2 x 6 x 24 5 x 3 - = - x = or x =

  7. Q15

    November 2023 · Paper 6

    5 marksA18 Quadratic equations
    The whole paper ↗
    November 2023 Paper 6, question 15: (a) Sasha and Taylor are asked to find how many solutions the equation ( 5 x 2 ) 45 has. + = Here is Sasha’s answer. Here is Taylor’s answer. 5( + 2)2 = 45 5( +

  8. Q4

    June 2023 · Paper 5

    3 marksG16 Area of 2D shapes and volume of prismsA18 Quadratic equations
    The whole paper ↗
    June 2023 Paper 5, question 4: The diagram shows a prism of length 10 cm. h cm 10 cm b cm The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the he

  9. Q15

    June 2023 · Paper 5

    9 marksA4 Simplifying, expanding and factorisingA18 Quadratic equations
    The whole paper ↗
    June 2023 Paper 5, question 15: (a) Factorise. 2 - 9 x 4 (a) (b) Solve by factorisation. 2 - 3 x 2 x 8 0 - = (b) x = or x = (c) Solve. 2 ( x 5 ) - 2 = 1 3 x - (c) x =

  10. Q7

    June 2023 · Paper 6

    4 marksA18 Quadratic equationsA11 Quadratic graphs and turning points
    The whole paper ↗
    June 2023 Paper 6, question 7: The diagram shows the graph of y 2 , where k is an integer. kx x = - + y 4 3 2 1 x 0 -1 1 2 3 4 -1 -2 -3 (a) Show that k 3 . = 2 (b) Use the graph to solve x 3

  11. Q18

    November 2022 · Paper 5

    8 marksA18 Quadratic equationsA11 Quadratic graphs and turning points
    The whole paper ↗
    November 2022 Paper 5, question 18: (a) By factorising, find the roots of y 18 77 . x x = + + (a) x = and x = 2 2 (b) (i) Write y 18 77 in the form ( ) . x x y x a b = + + = + - (b)(i) y = 2 (ii)

  12. Q21

    November 2022 · Paper 5

    9 marksA14 Graphs in real contextsA18 Quadratic equations
    The whole paper ↗
    November 2022 Paper 5, question 21: The graph of y 2 G x 6 . = G x 2 - y 10 9 8 7 6 5 4 3 2 1 x -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 1 (a) There are no values of x for which x . k = 2 - Find the val

  13. Q13

    June 2022 · Paper 5

    7 marksA20 IterationA18 Quadratic equations
    The whole paper ↗
    June 2022 Paper 5, question 13: The graph of y 6 2 is shown below. x x = + - 2 + The roots of the equation x 6 x 2 0 are at p and q. - = y x p q O (a) (i) Calculate y when x = 1. (a)(i) y = (i

  14. Q17

    June 2022 · Paper 5

    6 marksA21 Forming and solving equationsA18 Quadratic equations
    The whole paper ↗
    June 2022 Paper 5, question 17: Charlie is making some wooden frames. Charlie has a strip of wood 1.6 m long and 2 cm wide. Not to scale 2 cm 1.6 m Each frame will be made from four pieces of

  15. Q19

    June 2022 · Paper 6

    5 marksA18 Quadratic equationsA11 Quadratic graphs and turning points
    The whole paper ↗
    June 2022 Paper 6, question 19: (a) Write x 8 9 in the form x . x a b + - - (a) (b) Use your answer from part (a) to solve. 2 - 8 9 0 x x + = Give your answers in exact form. You must show you

  16. Q16

    November 2021 · Paper 6

    9 marksA18 Quadratic equationsA21 Forming and solving equations
    The whole paper ↗
    November 2021 Paper 6, question 16: The following kinematics formulas may be used in this question. v u at = + 1 2 s ut at = + 2 2 2 v u 2 as = + The initial velocity of a particle is 20 m/s. The

  17. Q15

    November 2020 · Paper 4

    6 marksA18 Quadratic equations
    The whole paper ↗
    November 2020 Paper 4, question 15: Here are two pieces of work. For each one, describe the error made and give the complete correct solution. (a) Question: Solve by factorisation. 3x2 – 2x – 5 =

  18. Q19

    November 2020 · Paper 5

    7 marksA11 Quadratic graphs and turning pointsA18 Quadratic equations
    The whole paper ↗
    November 2020 Paper 5, question 19: (a) Write x 10 22 in the form ( ) . x x a b + - - (a) 2 (b) Sketch the graph of y 10 22 . x x = - + Show clearly the coordinates of any turning points and the v

  19. Q19

    November 2020 · Paper 6

    7 marksA18 Quadratic equationsA11 Quadratic graphs and turning points
    The whole paper ↗
    November 2020 Paper 6, question 19: The graph of y 2 3 9 is drawn below. x x = + - y 11 10 9 8 7 6 5 4 3 2 1 x 0 -3 -2 -1 1 2 3 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 2 + (a) Use the graph to solve x 2 3

  20. Q6

    November 2019 · Paper 5

    3 marksA18 Quadratic equations
    The whole paper ↗
    November 2019 Paper 5, question 6: Solve by factorising. 2 + x 9 x 20 0 + = x = or x =
All 33 questions on Quadratic equations, as a list
  1. June 2025, Paper 6, Q9 (6 marks): (a) Solve. 8 x 11 6 x 3 - = + (a) x = (b) Solve by factorising. 2 + x 15 x 16 0 - = (b) x = or x =
  2. November 2024, Paper 5, Q18 (6 marks): Solve. 2 x 5 - 4 x = x 4 - You must show your working. x = or x =
  3. November 2024, Paper 6, Q20 (6 marks): ( 4 x a )( 4 x a x )( 2 ) 16 x bx 50 + - + = + - Find the two possible pairs of values for a and b. You must show your working. Pair 1: a = and b = Pair 2: a =
  4. June 2024, Paper 4, Q15 (10 marks): (a) Solve by factorisation. 2 + 3 x 10 x 8 0 - = (a) x = or x = 2 + 2 (b) Write x 8 x 11 in the form ( x a ) b . + + - (b) 2 (c) (i) Write down the coordinates
  5. June 2024, Paper 6, Q8 (6 marks): (a) Complete this table for y x 3 x . = - x 0 1 2 3 4 2 1 - - 0 16 y 20 2 4 - - - 3 2 (b) Draw the graph of y x 3 x for values of x from 2 - to 4. = - y 20 15 1
  6. November 2023, Paper 4, Q18 (4 marks): Solve this quadratic equation by factorisation. 2 - 2 x 6 x 24 5 x 3 - = - x = or x =
  7. November 2023, Paper 6, Q15 (5 marks): (a) Sasha and Taylor are asked to find how many solutions the equation ( 5 x 2 ) 45 has. + = Here is Sasha’s answer. Here is Taylor’s answer. 5( + 2)2 = 45 5( +
  8. June 2023, Paper 5, Q4 (3 marks): The diagram shows a prism of length 10 cm. h cm 10 cm b cm The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the he
  9. June 2023, Paper 5, Q15 (9 marks): (a) Factorise. 2 - 9 x 4 (a) (b) Solve by factorisation. 2 - 3 x 2 x 8 0 - = (b) x = or x = (c) Solve. 2 ( x 5 ) - 2 = 1 3 x - (c) x =
  10. June 2023, Paper 6, Q7 (4 marks): The diagram shows the graph of y 2 , where k is an integer. kx x = - + y 4 3 2 1 x 0 -1 1 2 3 4 -1 -2 -3 (a) Show that k 3 . = 2 (b) Use the graph to solve x 3
  11. November 2022, Paper 5, Q18 (8 marks): (a) By factorising, find the roots of y 18 77 . x x = + + (a) x = and x = 2 2 (b) (i) Write y 18 77 in the form ( ) . x x y x a b = + + = + - (b)(i) y = 2 (ii)
  12. November 2022, Paper 5, Q21 (9 marks): The graph of y 2 G x 6 . = G x 2 - y 10 9 8 7 6 5 4 3 2 1 x -2 -1 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 1 (a) There are no values of x for which x . k = 2 - Find the val
  13. June 2022, Paper 5, Q13 (7 marks): The graph of y 6 2 is shown below. x x = + - 2 + The roots of the equation x 6 x 2 0 are at p and q. - = y x p q O (a) (i) Calculate y when x = 1. (a)(i) y = (i
  14. June 2022, Paper 5, Q17 (6 marks): Charlie is making some wooden frames. Charlie has a strip of wood 1.6 m long and 2 cm wide. Not to scale 2 cm 1.6 m Each frame will be made from four pieces of
  15. June 2022, Paper 6, Q19 (5 marks): (a) Write x 8 9 in the form x . x a b + - - (a) (b) Use your answer from part (a) to solve. 2 - 8 9 0 x x + = Give your answers in exact form. You must show you
  16. November 2021, Paper 6, Q16 (9 marks): The following kinematics formulas may be used in this question. v u at = + 1 2 s ut at = + 2 2 2 v u 2 as = + The initial velocity of a particle is 20 m/s. The
  17. November 2020, Paper 4, Q15 (6 marks): Here are two pieces of work. For each one, describe the error made and give the complete correct solution. (a) Question: Solve by factorisation. 3x2 – 2x – 5 =
  18. November 2020, Paper 5, Q19 (7 marks): (a) Write x 10 22 in the form ( ) . x x a b + - - (a) 2 (b) Sketch the graph of y 10 22 . x x = - + Show clearly the coordinates of any turning points and the v
  19. November 2020, Paper 6, Q19 (7 marks): The graph of y 2 3 9 is drawn below. x x = + - y 11 10 9 8 7 6 5 4 3 2 1 x 0 -3 -2 -1 1 2 3 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 2 + (a) Use the graph to solve x 2 3
  20. November 2019, Paper 5, Q6 (3 marks): Solve by factorising. 2 + x 9 x 20 0 + = x = or x =
  21. November 2019, Paper 6, Q7 (10 marks): (a) Complete the table for y 4 + . 1 x x = - -1 x 0 1 2 3 4 5 -2 y 1 6 2 + for -1 (b) Draw the graph of y x 4 x 1 x 5 . = - G G y 8 6 4 2 x 0 -1 1 2 3 4 5 -2 -4
  22. June 2019, Paper 4, Q19 (4 marks): Solve this equation algebraically. Give your solutions correct to 2 decimal places. 2 + 3 x 8 x 5 0 - = x = or x =
  23. June 2019, Paper 6, Q15 (3 marks): Solve by factorisation. 2 + 5 x 7 x 2 0 + = x = or x =
  24. November 2018, Paper 4, Q10 (2 marks): Here is the graph of y 6 . x x = + - y 6 5 4 3 2 1 x –4 –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 –6 –7 2 + Use the graph to solve the equation x 6 0 . x - = x = or x =
  25. November 2018, Paper 4, Q16 (3 marks): Solve by factorisation. 2 - 2 x 19 x 33 0 - = x = or x =
  26. November 2018, Paper 5, Q18 (10 marks): (a) (i) Write x 4 16 in the form ( ) . x x a b - + - (a)(i) 2 + (ii) Solve the equation x 4 16 0 . x - = Give your answers in surd form as simply as possible. (
  27. June 2018, Paper 4, Q16 (3 marks): Solve by factorisation. 3x2 + 11x – 20 = 0 x = or x =
  28. June 2018, Paper 4, Q20 (6 marks): Solve this equation, giving your answers correct to 1 decimal place. 5 3 2 + = x 2 x 3 + - x = or x =
  29. June 2018, Paper 5, Q9 (5 marks): (a) Complete the table for y 3 . x x = - -3 -2 -1 x 0 1 2 3 -18 -2 -2 y 0 2 18 3 for -3 G x G 3. (b) Draw the graph of y x 3 x = - y 20 15 10 5 3 x 0 –3 –2 –1 1
  30. November 2017, Paper 4, Q13 (8 marks): (a) Solve. x2 - 6x + 15 = 3x - 5 (a) x = or x = (b) Expand and simplify. (2x - 1)(x + 5)(3x - 2) (b)
  31. November 2017, Paper 4, Q18 (4 marks): Solve this equation algebraically. Give your solutions correct to 2 decimal places. 3x2 + 5x - 1 = 0 x = or x =
  32. June 2017, Paper 4, Q18 (6 marks): (a) Solve by factorisation. 2 + 2 x 5 x 12 0 - = (a) x = or x = (b) Solve this equation. Give each value correct to 2 decimal places. 2 + 3 x 2 x 3 0 - = (b) x
  33. June 2017, Paper 6, Q7 (5 marks): (a) Complete the table for y x 2 x . = - -1 0 1 2 3 4 x -1 3 0 0 3 y for -1 2 (b) Draw the graph of y 2 4 . x x x = - G G y 10 9 8 7 6 5 4 3 2 1 x 0 -1 1 2 3 4

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