Turn any quadratic into clear factors or roots. Learn methods, from AC grouping to GCF. Download results quickly and keep your math notes organized.
Try these inputs to see common factoring patterns.
| Example | a, b, c | Factored form | Notes |
|---|---|---|---|
| Perfect square | 1, 6, 9 | (x + 3)(x + 3) | Δ = 0 gives a repeated root. |
| AC grouping | 2, 7, 3 | (2x + 1)(x + 3) | Split 7x into 6x + x. |
| Difference of squares | 1, 0, -16 | (x - 4)(x + 4) | Recognize x² − 4². |
| Not integer-factorable | 3, 2, 2 | 3(x - r₁)(x - r₂) | Use roots when no integers fit. |
Tip: For best integer factoring, use integer coefficients. If you enter decimals, the calculator rounds to nearby integers for the factoring attempt.
A quadratic can be expressed as a product of two binomials when integer pairs match its coefficients. When this happens, expanding the factors recreates ax² + bx + c. If no integer pair exists, the calculator still reports the discriminant and approximate roots so you can interpret intercepts and turning behavior.
The discriminant Δ = b² − 4ac controls the number of real roots. When Δ > 0, there are two distinct real roots and the graph crosses the x-axis twice. When Δ = 0, there is one repeated root and the graph touches the x-axis once. When Δ < 0, the roots are complex and the parabola does not cross the x-axis.
Pulling out a greatest common factor reduces the numbers you factor next and keeps signs consistent. Example: 6x² + 12x + 6 becomes 6(x² + 2x + 1), then factors to 6(x + 1)(x + 1). This step also clarifies scaling in the Plot section.
For a ≠ 1, the AC method searches integers m and n such that m + n = b and m·n = a·c. The middle term is split into mx + nx, then factoring by grouping produces two binomials. This is the same structure you see in the worked steps output.
A correct factorization predicts the x‑intercepts. If factors are (px + q)(rx + s), then roots occur where each factor equals zero. The Plot shows the curve on [−10, 10] and, when real, marks both roots at y = 0. Use this visual check to catch sign errors quickly.
CSV exports a compact summary for spreadsheets: input, factor form, Δ, and roots. PDF exports the same summary and optionally includes a trimmed step list for documentation. These downloads are generated instantly from your entered coefficients, helping you archive solutions consistently.
It means no integer binomial factors were found. You still get Δ and approximate roots, and a real-factor form is shown when Δ is non‑negative.
Δ tells whether a quadratic has two, one, or zero real roots. That directly predicts how many x‑intercepts the plotted parabola will have.
Factoring is attempted using integer coefficients. If you enter decimals, the calculator rounds to nearby integers for factoring, while still computing Δ and roots from those integers.
The plot uses x from −10 to 10 with 401 points. This range is wide enough for most classroom problems and keeps the graph responsive on small screens.
Expand the reported factors and confirm you return to ax² + bx + c. The Plot also helps: real roots should align with the x‑axis crossings.
Many quadratics have non‑integer roots. When no integer pairs satisfy the coefficient conditions, factoring over integers is impossible, even though real roots may exist.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.