Enter quadratic coefficients
ax² + bx + cResult
Enter values and press the button to see steps.
Formula used
X‑Method (AC Method). For ax² + bx + c: compute ac = a·c. Find integers m,n with m+n=b and m·n=ac. Rewrite as ax² + mx + nx + c, then factor by grouping using g1=gcd(a,m) and g2=gcd(n,c). If no such m,n exist, the trinomial is irreducible over integers. You may still factor over rationals/reals using a constant multiple and the quadratic formula.
Scaling decimals. When a,b,c include decimals, multiply by a power of ten to clear decimals, factor the integer trinomial, then divide overall by the same power. This preserves equality up to a nonzero constant factor.
How to use this calculator
- Enter a, b, c for ax² + bx + c. Decimals allowed.
- Press Factor using X‑Method to compute steps and factors.
- Open the steps panel to see scaling, pairs, grouping, and results.
- Download the steps and results as CSV or PDF using the buttons.
- If irreducible over integers, roots are shown for further analysis.
Example data table
| # | a | b | c | ac | m,n | Factors (normalized) |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 5 | 5 | — | Irreducible over ℤ |
| 2 | 2 | 7 | 3 | 6 | [1,6] | (2x + 1)(x + 3) |
| 3 | 3 | -2 | -5 | -15 | [3,-5] | (x + 1)(3x - 5) |
| 4 | 4 | 4 | -12 | -48 | — | Irreducible over ℤ |
| 5 | 5 | 1 | 1 | 5 | — | Irreducible over ℤ |
FAQs
It rewrites ax² + bx + c as ax² + mx + nx + c where m+n=b and m·n=ac, then factors by grouping.
Yes. The calculator clears decimals by scaling, factors the integer form, then restores a rational constant factor to match the original trinomial.
Then the trinomial is irreducible over integers. The tool shows exact roots using the quadratic formula and notes irreducibility over ℤ.
Factoring is defined up to a nonzero constant. After clearing decimals and normalizing, we reintroduce a rational constant so the product equals the original trinomial.
CSV is generated client‑side from the step log. PDF uses a lightweight client library to render the same steps and the resulting factorization.