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Extended Only Algebra and Graphs

Solving Quadratic Equations for IGCSE Maths

Solving by factorising, using the formula, or completing the square. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding solving quadr

What You Need to Know

Solving by factorising, using the formula, or completing the square. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding solving quadratic equations is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Solving Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0. There are three methods to solve: factorising (quickest when factors are obvious), the quadratic formula x = (−b ± √(b²−4ac))/2a (always works), and completing the square (needed for exact surd answers and for finding the vertex). The discriminant b²−4ac tells you the number of solutions: positive = two solutions, zero = one repeated solution, negative = no real solutions. Quadratics appear heavily in IGCSE Extended Paper 4.

Step-by-Step Method

  1. 1

    Rearrange to ax² + bx + c = 0

    All terms on one side, zero on the other. Ensure the coefficient of x² is positive (multiply through by −1 if needed).

  2. 2

    Try factorising first

    Find two numbers multiplying to ac and adding to b. If this works quickly, factorise and set each factor to zero.

  3. 3

    Use the quadratic formula if factorising is not obvious

    x = (−b ± √(b²−4ac)) / 2a. Identify a, b, c carefully — remember b can be negative.

  4. 4

    Calculate both solutions

    Work out x = (−b + √D)/2a and x = (−b − √D)/2a separately. Round to 3 s.f. unless exact answers are needed.

  5. 5

    Check if solutions are valid in context

    For real-world problems, negative solutions may be inadmissible (e.g. lengths must be positive).

Worked Example

Question

Solve 2x² − 5x − 3 = 0.

Solution

Method 1: Factorising. a×c = 2×(−3) = −6. Find two numbers adding to −5, multiplying to −6: −6 and +1. 2x² − 6x + x − 3 = 0 2x(x − 3) + 1(x − 3) = 0 (2x + 1)(x − 3) = 0 2x + 1 = 0 → x = −1/2 x − 3 = 0 → x = 3 Method 2 (verify with formula): a=2, b=−5, c=−3. b²−4ac = 25+24 = 49 x = (5 ± 7)/4 x = 3 or x = −0.5 ✓ Answers: x = 3 or x = −0.5

Exam Tips for Solving Quadratic Equations

  • The quadratic formula is on the IGCSE formula sheet — but you must read off a, b, c correctly, including signs.
  • Before using the formula, make sure the equation is in the form ax²+bx+c = 0 (not ax²+bx = d).
  • If the discriminant is negative, write 'no real solutions' — do not try to take the square root of a negative number.
  • Factorising is fastest when the coefficients are small integers — always try it first before reaching for the formula.

Practice Questions

Q1: Solve x² − 7x + 10 = 0 by factorising.

Show hint

Find two numbers adding to −7 and multiplying to 10: −5 and −2.

Q2: Solve 3x² + 2x − 8 = 0 using the quadratic formula. Give answers correct to 2 decimal places.

Show hint

a=3, b=2, c=−8. Discriminant = 4+96 = 100. x = (−2 ± 10)/6.

Q3: A rectangle has length (x+5) cm and width (x−1) cm. Its area is 24 cm². Form a quadratic equation and find the value of x.

Show hint

(x+5)(x−1) = 24 → x² + 4x − 5 = 24 → x² + 4x − 29 = 0. Only the positive solution is valid.

Frequently Asked Questions

What is solving quadratic equations in IGCSE Maths?

Solving by factorising, using the formula, or completing the square.

Is solving quadratic equations in the Core or Extended syllabus?

Solving Quadratic Equations is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise solving quadratic equations effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on solving quadratic equations rather than trying to cover everything at once.

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