Direct answer
Factoring rewrites a polynomial as a product; for a quadratic, real linear factors correspond to its real roots, and expanding the factors must reproduce every original coefficient.
What this calculation tells you
Factorisation exposes values that make a product zero and can simplify equations, rational expressions, and structural comparisons.
The coefficient domain matters: an expression may factor over integers, rationals, reals, or complex numbers but not a narrower set.
Where it is used
Equation solving
Use the zero-product property after moving all terms to one side.
Graph interpretation
Connect real factors with x-intercepts and repeated roots.
Symbolic simplification
Cancel only genuine common factors under their domain restrictions.
Education
Recognise common factors, difference of squares, and perfect-square patterns.
When this guide helps
- Factoring a quadratic trinomial.
- Checking whether roots are rational.
- Recognising a repeated factor.
- Expanding factors to verify the result.
Remove common factors first
A greatest common factor simplifies the remaining structure and must not be lost when solving or reconstructing the polynomial.
Use roots and the discriminant
A nonnegative discriminant permits real roots. Integer coefficients with a perfect-square discriminant can yield exact rational factors.
Respect the coefficient domain
An irreducible integer quadratic may still factor approximately over the reals or exactly over the complex numbers. Label the form honestly.
Common mistakes
Frequent errors include losing the leading coefficient, cancelling terms that are not factors, and presenting rounded roots as exact factors.
- Factor out the GCF.
- Expand to check.
- Label exact versus approximate.
Worked case: monic trinomial
Factor x² + 5x + 6.
Find numbers whose product is 6 and sum is 5: 2 and 3.
x² + 5x + 6 = (x+2)(x+3).
Expanding returns the original coefficients, providing a complete check.
Reproduce this worked caseOpen Factoring Methods Calculator
Worked case: non-monic expression
Factor 6x² + x - 12.
The pair (2x+3)(3x-4) expands to 6x² -8x +9x -12 = 6x² + x -12.
The factorization is (2x+3)(3x-4).
Checking both middle cross-products prevents a plausible but incorrect factor pair.
Reproduce this worked caseOpen Factoring Methods Calculator
Compare quadratic factoring cases before generalising
Not every quadratic factors over integers. Completing the square or the quadratic formula remains available when no suitable factor pair exists.
| Expression | Factors | Expansion check |
|---|---|---|
| x²+5x+6 | (x+2)(x+3) | x²+5x+6 |
| 6x²+x-12 | (2x+3)(3x-4) | 6x²+x-12 |
quadratic factoring: calculation checklist
- Remove common factor first
- Match leading and constant products
- Check cross terms
- Expand to verify
- State number system
Practical questions
Frequently asked questions
Does every quadratic factor?
Every nonconstant quadratic factors over the complex numbers, but not necessarily into integer, rational, or real linear factors.
How do factors relate to roots?
A factor x−r is zero at x=r, so r is a root of the polynomial.
Why expand the answer?
Expansion verifies the leading, middle, and constant coefficients independently.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.