IGCSE Math RevisionQuestion 1
Solve the equation \(x^2 - 9x + 20 = 0\) by factorising.
Show the worked answer
- Two numbers that multiply to \(20\) and add to \(-9\): \(-4\) and \(-5\).
- \((x - 4)(x - 5) = 0\)
Answer: \(x = 4\) or \(x = 5\)
Cambridge IGCSE Maths (0580) · Topic 2: Algebra and graphs · 2.5
Cambridge IGCSE Maths 0580 Quadratic Equations Questions
Quadratic Equations questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.
- 8 questions
- 8 Extended
- 6 non-calculator
- Free, no sign-in
What the syllabus asks
- 2.5 Equations Core Extended
Construct expressions, equations and formulas; Solve linear equations in one unknown; Solve simultaneous linear equations in two unknowns; Change the subject of simple formulas.
Extended also: Fractional equations with numerical and linear algebraic denominators; Simultaneous linear and non-linear equations; Quadratic equations by factorising, completing the square and the quadratic formula; Change the subject where the subject appears twice or as a power or root.
Summarised in our own words: check the exact wording in the official 0580 syllabus.
Quadratic Equations questions: Extended
Extended content, for Papers 2 and 4. Core students can skip these.
Question 2
Solve the equation \(3x^2 - 7x + 2 = 0\).
Show the worked answer
- \((3x - 1)(x - 2) = 0\)
- \(3x - 1 = 0\) gives \(x = \frac{1}{3}\); \(x - 2 = 0\) gives \(x = 2\).
Answer: \(x = \frac{1}{3}\) or \(x = 2\)
Question 3
Solve \(3x^2 - 5x - 4 = 0\). Give your answers correct to 2 decimal places.
Show the worked answer
- \(a = 3\), \(b = -5\), \(c = -4\).
- \(x = \frac{5 \pm \sqrt{25 + 48}}{6} = \frac{5 \pm \sqrt{73}}{6}\)
- \(x = 2.2573\ldots\) or \(x = -0.5906\ldots\)
Answer: \(x = 2.26\) or \(x = -0.59\)
Question 4
(a) Express \(x^2 + 10x + 16\) as \((x + a)^2 + b\).
(b) Use your answer to solve \(x^2 + 10x + 16 = 0\).
Show the worked answer
- (a) Half of \(10\) is \(5\): \((x + 5)^2 = x^2 + 10x + 25\), so \(x^2 + 10x + 16 = (x + 5)^2 - 9\).
- (b) \((x + 5)^2 = 9\), so \(x + 5 = \pm 3\).
Answer: (a) \((x + 5)^2 - 9\) (b) \(x = -2\) or \(x = -8\)
Question 5
Solve \(x^2 - 4x - 1 = 0\) by completing the square. Give your answers in surd form.
Show the worked answer
- \(x^2 - 4x = (x - 2)^2 - 4\), so \((x - 2)^2 - 5 = 0\).
- \((x - 2)^2 = 5\), so \(x - 2 = \pm\sqrt{5}\).
Answer: \(x = 2 + \sqrt{5}\) or \(x = 2 - \sqrt{5}\)
Question 6
Express \(3x^2 - 12x + 7\) as \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are integers.
Show the worked answer
- \(3x^2 - 12x + 7 = 3(x^2 - 4x) + 7\)
- \(x^2 - 4x = (x - 2)^2 - 4\)
- \(3((x - 2)^2 - 4) + 7 = 3(x - 2)^2 - 5\)
Answer: \(3(x - 2)^2 - 5\)
Question 7
A rectangle is \(x\) cm wide and \((x + 3)\) cm long. Its area is \(40\) cm².
Find the perimeter of the rectangle.
Show the worked answer
- \(x(x + 3) = 40\), so \(x^2 + 3x - 40 = 0\).
- \((x + 8)(x - 5) = 0\). A width cannot be negative, so \(x = 5\).
- The rectangle is \(5\) cm by \(8\) cm: perimeter \(= 26\) cm.
Answer: \(26\) cm
Question 8
The sides of a right-angled triangle are \(x\) cm, \((x + 7)\) cm and \((x + 8)\) cm, where \((x + 8)\) cm is the hypotenuse.
Find \(x\).
Show the worked answer
- Pythagoras: \(x^2 + (x + 7)^2 = (x + 8)^2\).
- \(2x^2 + 14x + 49 = x^2 + 16x + 64\), so \(x^2 - 2x - 15 = 0\).
- \((x - 5)(x + 3) = 0\); a length is positive, so \(x = 5\) (sides \(5\), \(12\), \(13\)).
Answer: \(x = 5\)
Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).
Keep going
- Algebra and graphs practice for 0580: more questions, each one marked, tagged Core or Extended
- My mistakes, then a Skill Builder on the skills you missed
- Common mistakes in 0580: what the Cambridge examiner reports say
- Algebra and graphs revision notes and the 0580 formula sheet
- Cambridge IGCSE Maths 0580 past papers, when you are ready for a full paper
Questions students ask
Which Cambridge 0580 papers can test quadratic equations?
Quadratic Equations is on both tiers (syllabus 2.5), so it can come up on Core Papers 1 (non-calculator) and 3 (calculator), and Extended Papers 2 (non-calculator) and 4 (calculator). Extended candidates also need the Extended parts listed on this page. Half of each tier's marks are on a non-calculator paper, so practise the non-calculator questions without one.
Are these Cambridge past-paper questions?
No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.
Where can I practise more quadratic equations questions?
In the 0580 practice for topic 2 (Algebra and graphs), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.