Factoring quadratic equations with steps
Use this page when you suspect the quadratic should factor into brackets and you want to see the route clearly before trying a different solving method.
Start here if you want the short version before reading the full method.
- Factoring a quadratic means rewriting it as two brackets and then setting each bracket equal to 0.
- It works best when the numbers produce a clean factor pair or a tidy grouping route.
What this topic means and what to look for first.
This page focuses on the factorising method only, so it is best when you already know the equation is quadratic and you want to see whether brackets will solve it efficiently.
The key question is not just whether the numbers multiply correctly, but whether the middle term is rebuilt correctly as well.
One reliable route through the topic.
- 1Write the equation in the form ax^2 + bx + c = 0.
- 2If a = 1, find two numbers that multiply to c and add to b.
- 3If a is not 1, use the ac method or factor by grouping instead of guessing brackets too early.
- 4Write the factorised brackets carefully and set each bracket equal to 0.
- 5Check the roots by substitution or by expanding the brackets again.
See the method in action.
x^2 + 5x + 6 = 0
- Look for two numbers that multiply to 6 and add to 5.
- Those numbers are 2 and 3, so the factorisation is (x + 2)(x + 3) = 0.
- Set each bracket equal to 0 to get x = -2 or x = -3.
x^2 - x - 12 = 0
- Look for two numbers that multiply to -12 and add to -1.
- Those numbers are -4 and 3.
- So the factorisation is (x - 4)(x + 3) = 0, giving x = 4 or x = -3.
2x^2 + 7x + 3 = 0
- Multiply a and c: 2 × 3 = 6, then find numbers that multiply to 6 and add to 7.
- Split the middle term: 2x^2 + 6x + x + 3 = 0.
- Factor by grouping to get (2x + 1)(x + 3) = 0, so x = -1/2 or x = -3.
x^2 + 4x + 1 = 0
- No integer factor pair multiplies to 1 and adds to 4.
- That is a sign to stop forcing brackets and switch to completing the square or the quadratic formula.
- This quadratic can still be solved, but not by a quick integer factor pair.
Things that commonly send the method off track.
- Using a pair that multiplies correctly but adds to the wrong middle term.
- Forgetting that one positive and one negative number may be needed.
- Ignoring the coefficient before x^2 and treating every quadratic as if it starts with x^2.
- Forcing brackets onto a quadratic that does not factor neatly over the integers.
Use a short verification pass before moving on.
- Expand the brackets to check that they rebuild the original quadratic exactly.
- Substitute each root into the original equation and confirm the result is 0.
Try a few variations before switching to a calculator or solver tool.
- x^2 + 8x + 15 = 0
- x^2 - 6x + 8 = 0
- 2x^2 + 9x + 4 = 0
- 3x^2 - x - 2 = 0
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Extra algebra revision resources
If you want more printed algebra practice after this page, these broader searches are a sensible next step.
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Algebra workbook and revision book search
Useful if you want more equation, factorising, and worked-example practice in one printed source.
View Algebra workbook and revision book searchAmazon
GCSE algebra practice resources search
A wider GCSE-style search if you want more mixed algebra questions beyond one online guide.
View GCSE algebra practice resources searchNeed live help with factoring quadratics?
If factor pairs or grouping still feel shaky, use the live-help route and mention the exact quadratic format you need help with.
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Short answers worth checking.
Write it in standard form, find the correct factor pair or grouping split, write the brackets, and then set each bracket equal to zero.
Switch to another method such as completing the square or the quadratic formula instead of forcing a bracket pattern that is not there.
Use the ac method or factor by grouping so the coefficient is handled properly instead of guessing too early.
Continue with the next closely related topic.
Use the public site structure first, then switch into the solver tool only if you need a direct test.
CureMath uses artificial intelligence to suggest how a maths problem could potentially be solved. AI can make mistakes.
Check important answers independently before relying on them.