GCSE MATHS • HIGHER TIER
Solving Quadratic Equations by Factorising Made Simple
Factorise quadratic equations, apply the Zero Product Rule and find both solutions accurately.
Clear method
Exam practice
Worked proofs
Solving Quadratics by Factorising in Four Moves
1
Rearrange
Make sure the quadratic equation is equal to zero.
2
Factorise
Rewrite the quadratic expression as a product of factors.
3
Set to Zero
Set each factor equal to zero using the Zero Product Rule.
4
Solve and Check
Solve each equation and check the solutions in the original quadratic.
What You’ll Learn
Recognise a quadratic equation.
Understand what solving by factorising means.
Use the Zero Product Rule.
Find the correct factor pairs.
Solve quadratic equations after factorising.
Handle quadratics with a common factor.
Check solutions by substitution.
Connect quadratic solutions with graph x-intercepts.
Key Rules
Make it equal zero
Rearrange the quadratic so it is written in the form **ax² + bx + c = 0
Find the factors
Choose two numbers that multiply to give c & add to give b, then factorise the quadratic.
Zero Product Rule
If (a)(b) = 0, then a = 0 or b = 0. Set each factor equal to zero.
Solve both factors
Solve each equation to find the solutions, then check by substituting them back into the original quadratic.
Worked Proof: Solving Quadratics by Factorising
Step-by-Step Method
- Rearrange the equation so it equals zero.
- Factorise the quadratic.
- Set each bracket equal to zero.
- Solve each equation.
- Check your answers.
More Worked Examples
1
Solve: x² + 7x + 12 = 0
Factorise:
(x + 3)(x + 4) = 0
2
Solve: x² - 9x + 20 = 0
Factorise:
(x - 4)(x - 5) = 0
3
Solve: x² + x - 12 = 0
Factorise:
(x + 4)(x - 3) = 0
4
Solve: x² - 2x - 15 = 0
Factorise:
(x - 5)(x + 3) = 0
- Common GCSE Mistakes
- Forgetting to make the equation equal zero first.
- Choosing incorrect factors.
- Forgetting there are usually two solutions.
- Making sign errors when factorising.
- Not checking answers.
- Exam Tips
- Make sure the equation equals zero before factorising.
- Write factor pairs systematically when the correct pair is not obvious.
- Check that your chosen factors multiply to the constant term and add to the x coefficient.
- Set each factor equal to zero separately.
- Remember to give both solutions when there are two.
Skills Used in Solving Quadratics by Factorising
Quadratic Equations
Factorising
Factor Pairs
Zero Product Rule
Equation Solving
Expanding Brackets
Ready to Practise?
Printable Worksheet
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Interactive Quiz
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Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Solving Quadratics by Factorising Tutorial
Step-by-step walkthroughs of Solving Quadratics by Factorising with clear methods and exam tips.
Frequently Asked Questions
It means rewriting the quadratic as factors and then setting each factor equal to zero.
The Zero Product Rule can be applied when the product of the factors equals zero.
Look for two numbers that multiply to give the constant term and add to give the coefficient of x.
A factorised quadratic usually gives two separate linear equations, one from each factor.
Take out the common factor first, then set each factor equal to zero.
Substitute each solution back into the original quadratic equation.