Quadratic formula practice questions

The quadratic formula solves any equation of the form ax squared plus bx plus c equals 0, giving x equals minus b plus or minus the square root of b squared minus 4ac, all over 2a. Work out the discriminant first, then substitute carefully using brackets around any negative values.

The quadratic formula solves any quadratic equation, including those that will not factorise. For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a.

The method, step by step

1. Rearrange to equal zero. The formula only works when one side is 0.
2. Identify a, b and c. Include the signs. A common slip is dropping a negative.
3. Work out the discriminant first. Calculate b² − 4ac before anything else.
4. Substitute carefully. Use brackets around negative values.
5. Split into two answers. The ± gives one solution using + and one using −.

Worked example

Common mistakes

Quadratic formula exam-style questions

Untimed practice with worked answers. Every answer is a whole number. This extra activity does not count towards your daily progress.

1. Solve x² − 7x + 12 = 0. Give the larger solution.
   Worked answer for question 1

   a = 1, b = −7, c = 12. b² − 4ac = 49 − 48 = 1, so x = (7 ± 1)/2, giving x = 4 or x = 3. The larger is 4.

2. Solve x² − 2x − 24 = 0. Give the positive solution.
   Worked answer for question 2

   b² − 4ac = 4 + 96 = 100, so x = (2 ± 10)/2, giving x = 6 or x = −4. The positive solution is 6.

3. Solve 2x² − 11x + 5 = 0. Give the larger solution.
   Worked answer for question 3

   a = 2, b = −11, c = 5. b² − 4ac = 121 − 40 = 81, so x = (11 ± 9)/4, giving x = 5 or x = 0.5. The larger is 5.

4. Find the value of the discriminant b² − 4ac for x² + 6x + 4 = 0.
   Worked answer for question 4

   b² − 4ac = 6² − 4 × 1 × 4 = 36 − 16 = 20. It is positive, so there are two real solutions.

5. x² − 14x + 49 = 0 has one repeated solution. What is it?
   Worked answer for question 5

   b² − 4ac = 196 − 196 = 0, so x = 14/2 = 7. The quadratic is (x − 7)².

Continue with GCSE daily practice

Common questions

Is the quadratic formula on the GCSE formula sheet?

Yes, on the Higher tier sheet only: Foundation students do not need it. See the GCSE Maths formula sheet for everything that is given on each tier.

When should I use the quadratic formula instead of factorising?

Use it when the quadratic does not factorise easily, or when the question asks for answers to decimal places or significant figures. That is a strong hint the solutions are not whole numbers.

What does the discriminant tell you?

The discriminant b² − 4ac tells you how many real solutions there are: two if it is positive, one repeated solution if it is zero, and none if it is negative.

Check your answers

Use the free quadratic equation solver to see full working for any quadratic: discriminant, factorising, the formula, exact surd answers and the turning point.

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More practice

Daily practice for Year 11GCSE flashcards (PDF)All free resources

Printable worksheets for this topic

Free A4 PDFs with worked answers. See every printable, Year 1 to A-level.

• GCSE Higher maths worksheetGCSE Higher · 6 A4 pages · Worked answer keyDownloadPrint
• A-level maths worksheetA-level Maths · 8 A4 pages · Worked answer keyDownloadPrint