Factorising Quadratics Generator
Expressions to factorise, built by multiplying two brackets together and then hiding them. A quadratic taken at random usually does not factorise over the integers at all, which is why practice questions have to be constructed backwards like this. The working shown is the split-the-middle-term method, which is the one that keeps working when the x squared coefficient is not one.
What this generator does
Draws two linear brackets, multiplies them out, and shows only the result. The answer is the brackets it started from, and the working gives the two numbers that multiply to ac and add to b — the pair that makes splitting the middle term work.
How to use this tool
- Choose how many expressions you want and which level to work at.
- Factorise each into two brackets.
- Reveal the answers to compare against your brackets.
- Show the working to see the pair of numbers used to split the middle term.
Understanding the controls
- How many questions
- Between 1 and 20. Ten is a normal homework set; fewer if the leading coefficient is not one.
- Level
- Gentle and standard keep the x squared coefficient at one so the factors can be spotted by inspection. Stretch puts a coefficient in front, which forces the grouping method.
- Seed
- Reproduces the same expressions exactly from a word you choose.
- Show the answers / working
- The factorised form, or the product-and-sum search that finds it.
Common use cases
- Practice where every expression definitely factorises
- Moving from a leading coefficient of one to a leading coefficient that is not
- Teaching the product-and-sum search with numbers small enough to hold in your head
- Homework that pairs with the expanding generator as its inverse
- Reissuing exactly the same ten questions to a group
How this generator works
Two brackets (ax + b)(cx + d) are drawn and expanded to ac x² + (ad + bc) x + bd, which is the expression you see. The check multiplies the stated factors back out and requires all three coefficients to match, so a question can never show factors that are not actually its factors.
Randomness and fairness
The hidden brackets are random; that the expression factorises follows from how it was constructed. Seeded sets are reproducible and are therefore explicitly not cryptographically secure. Without a seed the browser's cryptographically secure generator is used.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Every expression factorises into two linear brackets. Difference-of-two-squares and non-factorising cases are not distinguished as their own types.
- There is no common factor to take out first, which is a step some syllabuses teach before this one.
- Only quadratics — cubics and higher are not offered.
- Your working is not checked; the answer is revealed for you to compare.
- Sets are not kept between visits.
Privacy and your data
The brackets are drawn and expanded entirely in your browser. Nothing about the set is transmitted or stored.
Related generators
- Expanding Brackets Question GeneratorProducts of two linear brackets to multiply out, with each of the four partial products written separately.
- Quadratic Equation GeneratorQuadratics built from their roots, so they always factorise, with the discriminant stated and the repeated-root case named.
- Prime Factorisation Question GeneratorPrime-factorisation questions built to order, answered in index form, with the factors checked to be prime.
- Index Laws Question GeneratorSimplification questions on multiplying and dividing powers, in one letter or two, with the numbers and the indices handled separately.