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GCSE quadratic equations practice and worked examples

Factorise a quadratic, use the zero-product rule and check both solutions. This set focuses on factorisable equations, with a Higher extension using the quadratic formula.

Foundation and Higher · Useful near the top of Foundation for simple factorisable equations; Higher students can continue to formula questions.

Core pathway. Core: simple factorisable quadratics. Use the zero-product rule and keep both possible solutions.

Higher extension. Higher: the quadratic formula, including quadratics that do not factorise neatly. The formula section and its linked questions are labelled Higher.

2 worked examples and 8 original practice questions · Allow 25–40 minutes · Read online or print. Higher extensions are labelled. No account or email needed.

Learn the methodTry the questions

Printing includes the method, examples and space for working, followed by a separate answer section.

A useful approach

  1. Rearrange the equation so one side is zero.
  2. Look for a common factor or two brackets. A product is zero when at least one factor is zero.
  3. Solve each factor separately and check the roots; in a context, reject any root that is not possible.

A mistake to watch for

Dividing by x may lose the solution x = 0. Take out a factor of x and solve both factors instead.

Learn each method, then practise it

Read each line of working and explain why it follows from the previous line.

Solve by factorising

Rearrange to zero. If a product of two factors is zero, at least one factor is zero. Solve each factor equation separately.

Worked example 1

Solving Quadratics · Core GCSE skill · No calculator · 3 marks

Solve  x2 + 8x + 12 = 0

  1. Find two numbers that multiply to 12 and add to 8: they are 2 and 6
  2. Factorise: (x + 2)(x + 6) = 0
  3. So x = -2 or x = -6
Answer: x = -2 or x = -6
Report a problem with this question

Now practise this method: Question 1 · Question 2 · Question 3 · Question 4

Use the quadratic formula (Higher)

When factorising is not straightforward, write ax² + bx + c = 0 with a non-zero. The formula gives both possible roots using the plus and minus branches.

Why and when to use the quadratic formula · Higher

For ax² + bx + c = 0 with a ≠ 0, completing the square and rearranging gives the formula below. It works even when integer factors are not available. If the question specifies a method, use that method.

x=−b±b2−4ac2a
  1. Rearrange to zero and identify a, b and c, including each sign.
  2. Calculate the discriminant, b² − 4ac. Positive gives two real roots; zero gives one repeated real root; negative gives no real roots.
  3. Calculate one result with +√ and another with −√. The entire numerator is divided by 2a.
  4. Keep full calculator precision until the final rounding, and substitute each result to check.

In the worked formula example, a = 1, b = 5 and c = 3, so the discriminant is 25 − 12 = 13. The plus branch gives (−5 + √13)/2 ≈ −0.70; the minus branch gives (−5 − √13)/2 ≈ −4.30, both to 2 decimal places. The exact roots add to −5 and multiply to 3, matching the original quadratic.

Worked example 2

Quadratic Formula · Higher extension · Calculator allowed · 3 marks

Solve x2 + 5x + 3 = 0

Give your solutions correct to 2 decimal places.

  1. Use the quadratic formula with a = 1, b = 5, c = 3.
  2. x = = .
  3. √13 = 3.60555..., so x = = -0.69722... or x = = -4.30277...
  4. To 2 decimal places, x = -0.70 or x = -4.30.
Answer: x = -0.70, x = -4.30
Report a problem with this question

Now practise this method: Question 5 · Question 6 · Question 7 · Question 8

Your practice questions

Write your working on paper. Marks indicate how much working to show; these questions are self-marked and do not change saved practice results. Use squared paper for drawing questions.

Worked answers: Quadratic equations

Compare the reasoning as well as the final answer. Another correct method is valid. If a step is unclear, revisit an example before trying a similar question.

Answer 1

Show answer and working for question 1
  1. If a product of two brackets is 0, one of the brackets must equal 0
  2. x + 3 = 0 gives x = -3
  3. x - 9 = 0 gives x = 9
Answer: x = -3 or x = 9
Back to question 1

Answer 2

Show answer and working for question 2
  1. Find two numbers that multiply to -24 and add to -5: they are -8 and 3
  2. Factorise: (x - 8)(x + 3) = 0
  3. So x = 8 or x = -3
Answer: x = 8 or x = -3
Back to question 2

Answer 3

Show answer and working for question 3
  1. Find two numbers that multiply to 28 and add to -11: they are -4 and -7
  2. Factorise: (x - 4)(x - 7) = 0
  3. So x = 4 or x = 7
Answer: x = 4 or x = 7
Back to question 3

Answer 4

Show answer and working for question 4
  1. Let the smaller number be n, so the larger number is n + 5
  2. Then n(n + 5) = 84, so n2 + 5n - 84 = 0
  3. Find two numbers that multiply to -84 and add to 5: they are 12 and -7
  4. Factorise: (n + 12)(n - 7) = 0, so n = -12 or n = 7
  5. The numbers are positive, so n = 7 and the numbers are 7 and 12
  6. Check: 12 - 7 = 5 and 7 × 12 = 84 ✓
Answer: 7 and 12
Back to question 4

Answer 5

Show answer and working for question 5
  1. Use the quadratic formula with a = 1, b = -4, c = -9.
  2. x = = .
  3. √52 = 7.21110..., so x = = 5.60555... or x = = -1.60555...
  4. To 2 decimal places, x = 5.61 or x = -1.61. Take care with the signs: -b = 4 and -4ac = +36.
Answer: x = 5.61, x = -1.61
Back to question 5

Answer 6

Show answer and working for question 6
  1. Use the quadratic formula with a = 2, b = 7, c = -3.
  2. x = = .
  3. √73 = 8.54400..., so x = = 0.38600... or x = = -3.88600...
  4. To 2 decimal places, x = 0.39 or x = -3.89.
Answer: x = 0.39, x = -3.89
Back to question 6

Answer 7

Show answer and working for question 7
  1. Use the quadratic formula with a = 3, b = -5, c = -7.
  2. x = = .
  3. √109 = 10.44030..., so x = = 2.57338... or x = = -0.90671...
  4. To 2 decimal places, x = 2.57 or x = -0.91.
Answer: x = 2.57, x = -0.91
Back to question 7

Answer 8

Show answer and working for question 8
  1. Use the quadratic formula with a = 1, b = 6, c = 4.
  2. x = = .
  3. Simplify the surd: √20 = 2√5.
  4. x = = -3 ± √5.
Answer: x = -3 + √(5), x = -3 - √(5)
Back to question 8

Practise a specific skill

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Use this in class

Give students time to write the method, then compare their working with the answer section. Ask which step they would check first if their answer differs.

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