Quadratic Equation Generator
Quadratics are generated from a pair of roots and multiplied out, which is why every one of them factorises over the integers. That is the property a practice question needs and a randomly-chosen set of coefficients almost never has. The discriminant is reported alongside, because it is the number that tells you what kind of answer to expect before you find it, and because a repeated root — where the discriminant is zero — is easy to mistake for a mistake.
What this generator does
Picks two roots, multiplies out a(x − p)(x − q), and presents the result as an equation to solve. Because the roots came first the answer is exact, the discriminant is a perfect square, and the factorised form is known rather than searched for.
How to use this tool
- Choose how many quadratics you want and how large the roots may be.
- Solve each by factorising, or by the formula if you would rather practise that.
- Reveal the answers to check the roots you found.
- Show the working to see the discriminant and the factorised form.
Understanding the controls
- How many questions
- Between 1 and 20. Five is a reasonable set when each one is being solved two ways, by factorising and by formula.
- Level
- Gentle and standard keep the x squared coefficient at one, which is the case that factorises by inspection. Stretch allows a leading coefficient, which needs grouping or the formula.
- Seed
- Reproduces the same set of quadratics from the same word.
- Show the answers / working
- The roots alone, or the discriminant and the factorised form as well.
Common use cases
- Practice at factorising where every question is guaranteed to factorise
- Drilling the quadratic formula against answers you can verify by inspection
- Showing what a zero discriminant looks like as an equation rather than as a rule
- Stretch questions where the x squared coefficient is not one
- Generating a set of equations to test a solver against
How this generator works
Two roots p and q are drawn, and a leading coefficient a at the stretch level. The equation is a(x − p)(x − q) = 0 expanded to ax² + bx + c. The check substitutes each stated root back into those coefficients using exact fractions and confirms the result is exactly zero, and recomputes b² − 4ac against the discriminant shown.
Randomness and fairness
The roots and any leading coefficient are random; that the equation factorises is a property of how it was built, not of chance. A seeded set is reproducible and therefore explicitly not cryptographically secure. Unseeded, the browser's cryptographically secure generator supplies the draws.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Every quadratic here has whole-number roots. Irrational and complex roots never appear, so this does not practise surd form.
- Roots are drawn independently, so a repeated root turns up only occasionally rather than on demand.
- The question is always set equal to zero. Rearranging from a form like x² = 5x − 6 is a separate skill this does not cover.
- There is no graph. The link between roots and where a curve crosses is described, not drawn.
- A generated set is not saved, so seed one before you print it.
Privacy and your data
Every equation is built in your browser from your own choices. Nothing about the set you generate is sent anywhere or retained.
Related generators
- Factorising Quadratics GeneratorQuadratic expressions that are guaranteed to factorise, with the split-the-middle-term working shown.
- Expanding Brackets Question GeneratorProducts of two linear brackets to multiply out, with each of the four partial products written separately.
- Simultaneous Equations GeneratorPairs of linear equations built backwards from a whole-number solution, with the elimination worked out line by line.
- Simple Equation GeneratorLinear equations built to order — one step, two steps, unknowns on both sides and brackets — each with a whole-number solution.