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Quadratic Functions and Equations in One Variable · Form 4

Quadratic Functions and Equations in One Variable: Practice Questions

Original SPM-style practice questions for Quadratic Functions and Equations in One Variable, each with full worked solutions, Paper 1 multiple-choice and Paper 2 structured.

Written by the spmmath.com.my editorial team.

Original practice questions for Quadratic Functions and Equations in One Variable, in the style of Mathematics Paper 1 (Objective) and Mathematics Paper 2 (Subjective). Each answer is worked so you can check your method, not just the result.

Multiple-choice (Paper 1 style)

Question 1

Express (x − 3)(x + 5) = 7 in the general form ax² + bx + c = 0.

  1. A. x² + 2x − 22 = 0
  2. B. x² + 2x − 8 = 0
  3. C. x² − 2x − 22 = 0
  4. D. x² + 2x + 22 = 0

Question 2

Solve the quadratic equation 2x² − 7x + 3 = 0 by factorisation.

  1. A. x = ½ or x = 3
  2. B. x = −½ or x = −3
  3. C. x = ½ or x = −3
  4. D. x = 2 or x = 3

Question 3

A quadratic equation has roots −2 and 5. Which equation has these roots?

  1. A. x² − 3x − 10 = 0
  2. B. x² + 3x − 10 = 0
  3. C. x² − 3x + 10 = 0
  4. D. x² + 7x − 10 = 0

Question 4

Determine the number of real roots of the equation x² − 4x + 7 = 0.

  1. A. Two different real roots
  2. B. Two equal real roots
  3. C. No real roots
  4. D. Three real roots

Question 5

The equation x² + px + 9 = 0 has two equal roots. Find the positive value of p.

  1. A. 3
  2. B. 6
  3. C. 9
  4. D. 18

Question 6

Find the equation of the axis of symmetry of the graph y = x² − 6x + 5.

  1. A. x = −3
  2. B. x = 3
  3. C. x = 6
  4. D. x = 5

Question 7

For the quadratic function y = −2x² + 3x + 1, which statement about its graph is correct?

  1. A. Opens upward with a minimum point
  2. B. Opens downward with a maximum point
  3. C. Opens upward with a maximum point
  4. D. Opens downward with a minimum point

Question 8

At which points does the curve y = x² − 2x − 8 cut the x-axis?

  1. A. (4, 0) and (−2, 0)
  2. B. (−4, 0) and (2, 0)
  3. C. (8, 0) and (−1, 0)
  4. D. (2, 0) and (4, 0)

Question 9

Express x² + 8x + 3 in the form (x + p)² + q.

  1. A. (x + 4)² − 13
  2. B. (x + 4)² + 13
  3. C. (x + 8)² − 13
  4. D. (x + 4)² − 19

Question 10

Solve x² + 3x − 1 = 0 using the quadratic formula, giving the roots in surd form.

  1. A. (−3 ± √13)/2
  2. B. (3 ± √13)/2
  3. C. (−3 ± √5)/2
  4. D. (−3 ± √13)/1

Structured (Paper 2 style)

Question 1 (6 marks)

A rectangular vegetable plot has a length of (x + 4) metres and a width of (x − 1) metres. The area of the plot is 24 m².

(a) Show that x² + 3x − 28 = 0. (b) Solve the equation to find the value of x.

(c) Hence, state the length of the plot.

  1. (a) Area = length × width, so (x + 4)(x − 1) = 24.
  2. Expanding the left side: x² + 3x − 4 = 24.
  3. Moving 24 to the left: x² + 3x − 28 = 0. (shown)
  4. (b) Factorise: (x + 7)(x − 4) = 0.
  5. So x + 7 = 0 or x − 4 = 0, giving x = −7 or x = 4.
  6. The width (x − 1) must be positive, so reject x = −7; therefore x = 4.
  7. (c) Length = x + 4 = 4 + 4 = 8 m.

Question 2 (6 marks)

A quadratic function is given by f(x) = x² − 8x + 11. (a) Express f(x) in the form (x − h)² + k.

(b) State the coordinates of the minimum point. (c) State the equation of the axis of symmetry.

(d) Solve f(x) = 0, giving your answers correct to 2 decimal places.

  1. (a) Take half of −8 and square it: (−4)² = 16.
  2. f(x) = x² − 8x + 11 = (x − 4)² − 16 + 11 = (x − 4)² − 5.
  3. (b) The minimum value is −5 when x = 4, so the minimum point is (4, −5).
  4. (c) The axis of symmetry passes through the vertex: x = 4.
  5. (d) Set (x − 4)² − 5 = 0, so (x − 4)² = 5.
  6. Take square roots: x − 4 = ±√5, so x = 4 ± √5.
  7. x = 4 + 2.2361 = 6.24 or x = 4 − 2.2361 = 1.76 (2 d.p.).

Question 3 (7 marks)

The quadratic equation 2x² + (k + 1)x + 8 = 0 has two equal roots, where k > 0. (a) Using the discriminant, form an equation in k.

(b) Find the value of k. (c) Hence, solve the quadratic equation.

  1. (a) For two equal roots, b² − 4ac = 0, with a = 2, b = k + 1, c = 8.
  2. So (k + 1)² − 4(2)(8) = 0, that is (k + 1)² − 64 = 0.
  3. (b) (k + 1)² = 64, so k + 1 = ±8, giving k = 7 or k = −9.
  4. Since k > 0, k = 7.
  5. (c) Substitute k = 7: 2x² + 8x + 8 = 0.
  6. Divide by 2: x² + 4x + 4 = 0, so (x + 2)² = 0.
  7. Therefore x = −2 (repeated root).

Question 4 (8 marks)

A ball is thrown upward. Its height, h metres, above the ground after t seconds is modelled by h = 20t − 5t².

(a) Find the height of the ball after 1 second. (b) Find the times when the ball is at a height of 15 m.

(c) By completing the square, find the maximum height reached and the time it occurs. (d) Find the time when the ball hits the ground.

  1. (a) Substitute t = 1: h = 20(1) − 5(1)² = 20 − 5 = 15 m.
  2. (b) Set h = 15: 20t − 5t² = 15, so −5t² + 20t − 15 = 0.
  3. Divide by −5: t² − 4t + 3 = 0, then factorise: (t − 1)(t − 3) = 0.
  4. So t = 1 s or t = 3 s.
  5. (c) h = 20t − 5t² = −5(t² − 4t) = −5[(t − 2)² − 4] = −5(t − 2)² + 20.
  6. The maximum height is 20 m, reached when t = 2 s.
  7. (d) Set h = 0: 20t − 5t² = 0, so 5t(4 − t) = 0, giving t = 0 or t = 4.
  8. The ball hits the ground at t = 4 s (t = 0 is the moment of throwing).

Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)

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Frequently asked questions

What's the difference between the Paper 1 MCQ set and the Paper 2 structured set for this chapter?

The Paper 1 set gives quick multiple-choice questions on quadratic functions and equations to build speed and accuracy, while the Paper 2 set gives longer structured questions that need full working, similar to how marks are awarded in the real exam. Practise both, since SPM tests this chapter in each paper.

Why should I attempt each question myself before checking the worked solution?

Working through a question first, even if you get stuck, shows you exactly where your method breaks down, checking the solution too early just teaches you to recognise it, not solve it. Attempt every step you can, write down where you're unsure, then compare your working line by line against the solution.

What are common mistakes students make in this quadratic functions and equations practice set?

Common slips include mixing up the signs when factorising or using the formula, forgetting to complete the square correctly, and misreading whether a question asks for the roots, the turning point, or the axis of symmetry. Read each question carefully and check your answer fits what was actually asked, not just any solution to the equation.

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