Calculator inputs
Recent calculations
Formula used
General quadratic: ax2 + bx + c
AC product: a × c
Discriminant: D = b2 − 4ac
Quadratic formula: x = (−b ± √D) / 2a
The AC method searches for two integers whose product equals ac. Their sum must equal the middle coefficient b. This makes each transformation easier to inspect and trust.
A perfect-square discriminant produces rational roots and rational linear factors. A positive nonsquare discriminant gives two irrational real roots. Negative discriminants require complex roots for complete linear factoring.
How to use this calculator
- Choose separate coefficients or complete expression input.
- Enter a nonzero leading coefficient and remaining values.
- Select the factoring domain and desired precision.
- Choose which detailed result sections should appear.
- Press “Factor quadratic” to calculate and verify.
- Copy, share, print, or export the finished result.
Fractions, decimals, negatives, and missing terms are supported. Expression mode accepts common squared-variable notation and flexible spacing. Clear validation messages explain unsupported or incomplete mathematical input.
Worked example data
| Quadratic | Factored form | Classification |
|---|---|---|
| 6x² + 11x + 3 | (3x + 1)(2x + 3) | Integer factors |
| 8x² − 2x − 3 | (4x − 3)(2x + 1) | Integer factors |
| 12x² + 11x − 5 | (3x − 1)(4x + 5) | Integer factors |
| 10x² − 25x | 5x(2x − 5) | Common factor |
| 4x² + 12x + 9 | (2x + 3)² | Perfect square |
| 6x² + x + 1 | Irreducible over reals | Complex roots |
Frequently asked questions
What is a non-unit leading coefficient?
It is a leading coefficient whose absolute value exceeds one. The AC method handles this extra factoring difficulty efficiently. It separates the middle term before grouping common binomials.
Why multiply a and c?
The product guides the search for the split terms. Their sum must still match the original middle coefficient. This preserves equivalence while preparing terms for useful grouping.
Can the calculator use fractional coefficients?
Yes, fractions and terminating decimals are converted exactly. Common denominators create an equivalent integer-coefficient quadratic first. The outside scalar remains visible in the final factorization.
What happens when the quadratic will not factor?
The calculator checks the discriminant and selected number domain. It then reports irrational, complex, or irreducible behavior. Exact roots remain available through the quadratic formula.
How are negative leading coefficients handled?
A negative sign is moved into the outside scalar. The primitive quadratic then uses a positive leading coefficient. This keeps the grouping process clearer and more consistent.
Can it factor a quadratic with c equal to zero?
Yes, zero constants produce an immediate variable factor. The remaining linear factor is then simplified exactly. Roots include zero and the second rational solution.
How does verification work?
Integer binomial factors are multiplied back together automatically. The resulting coefficients are compared with the primitive quadratic. Matching coefficients confirm the factorization without numerical rounding.
What does a perfect-square discriminant mean?
It means the square root of D is rational. Therefore both roots are rational when coefficients are rational. Complete linear factoring is then possible over rational numbers.
Can I share a calculation?
Yes, the calculator uses query parameters in the URL. The copy-link button preserves every selected input and option. Recipients can reopen the same completed calculation directly online.