Factoring & Quadratic Trainer
Practice factoring, difference of squares and completing the square — including the quadratics that do not factor, which most of them do not.
Why "does not factor" is an answer
- Most quadratics do not factor over the integers. Only about 7% of those with small integer coefficients do — the rest have roots that are irrational or complex, and no amount of staring finds brackets that are not there.
- Textbook exercise sets hide that, because they are built backwards: pick two whole-number roots, multiply out, set the result as the question. Every problem factors, so alongside the method students absorb a false expectation — that failing to find the brackets means not trying hard enough.
- So the factoring questions here are NOT built backwards. Coefficients are drawn freely about half the time and whether it factors is whatever the arithmetic decides, which means "does not factor" comes up and is a correct answer worth typing.
- The test for it is exact: a quadratic factors over the integers precisely when its discriminant is a perfect square. That is quicker than trying pairs, and it is the thing actually worth learning from this drill.
- Answers are marked against the maths, not against a string. A factorisation is multiplied back out and compared with the question, so the brackets in either order both count; roots are checked by substituting them in. Writing a right answer in an unexpected way does not make it wrong.
How to use
- Pick which topics you want to practise.
- Answer, then read why — including when the answer is that it does not factor.
- Keep a streak going; the global board ranks the longest.
Frequently asked questions
Why do some questions not factor at all?
Because most quadratics do not. Only about 7% of those with small integer coefficients factor over the integers — the rest have roots that are irrational or complex, and no amount of staring finds brackets that are not there. Recognising that quickly is a real skill, and a drill that never shows the case cannot teach it.
Is that not how textbooks do it?
No, and that is the problem. Textbook exercise sets are built backwards: pick two whole-number roots, multiply out, set the result as the question. Every problem factors, so alongside the method students absorb a false expectation — that failing to find the brackets means not trying hard enough.
How do I tell quickly whether one factors?
Work out the discriminant, b squared minus 4ac. It factors over the integers exactly when that is a perfect square. It is far quicker than trying pairs of factors, and it is arguably the most useful thing this drill teaches.
Does it matter which order I write the brackets?
No. Answers are marked against the maths rather than against a string: a factorisation is multiplied back out and compared with the question, so either bracket order counts, and spacing and an explicit times sign are both fine. Roots may be given in any order too.
Could it mark a right answer wrong?
It is built hard against that. Every generated question is verified against an independent calculation before it is asked — 20,000 questions across all five kinds were put through that before shipping, checking factorisations by multiplying them out and roots by substituting them in.
What is the global board ranking?
The longest unbroken run of correct answers. Which topics you selected is not recorded, so a run of discriminant questions and a run of factorisations sit on the same table — treat it as a bit of fun rather than a like-for-like ranking.
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