Enter trinomial coefficients
Example data table
| Expression | Integer factorization | Note |
|---|---|---|
| -2x² - 7x + 4 | -(2x - 1)(x + 4) | Factor out -1 first, then group. |
| -x² + 5x - 6 | -(x - 2)(x - 3) | Negative leading term becomes easier after sign extraction. |
| -3x² + x + 2 | -(3x + 2)(x - 1) | Use a×c and split the middle term. |
| -4x² - 4x - 1 | -(2x + 1)² | Perfect-square trinomial after normalization. |
Formula used
General form: ax² + bx + c
Common factor rule: Take out the greatest common factor from a, b, and c before deeper factoring.
Negative leading term rule: If the leading coefficient is negative, factor out -1 so the inside trinomial starts with a positive square term.
AC method: For the reduced trinomial, compute a × c. Find two integers m and n such that m + n = b and mn = ac.
Quadratic discriminant: D = b² - 4ac
Quadratic roots: x = (-b ± √D) / (2a)
Vertex: x = -b / (2a), then substitute back to get y.
How to use this calculator
- Enter the coefficients a, b, and c from your trinomial.
- Pick a variable symbol and preferred decimal precision.
- Choose the graph window if you want a tighter plot.
- Press Factor Trinomial to calculate instantly.
- Read the factored form, discriminant, roots, and vertex.
- Review the stepwise method to understand the algebra.
- Use the graph to verify intercepts and turning behavior.
- Download CSV or PDF for study notes or sharing.
FAQs
1) Why do negative trinomials often start with factoring out -1?
Factoring out -1 makes the leading coefficient positive. That usually reveals cleaner factor pairs and makes the AC method easier to apply without sign confusion.
2) What does the AC method do here?
It multiplies the reduced leading coefficient and constant term. Then it searches for two numbers whose product matches that result and whose sum matches the middle coefficient.
3) Can every negative trinomial be factored over integers?
No. Some trinomials have irrational or complex roots, so they cannot be written as integer binomial factors. The calculator reports that case and still shows root information.
4) Why is the discriminant included?
The discriminant tells you the root type. Positive values give two real roots, zero gives a repeated root, and negative values produce complex roots.
5) What does the graph add to factoring practice?
The graph helps you see x-intercepts, turning behavior, and whether the sign pattern makes sense. It is a visual check for the algebraic answer.
6) Does the calculator handle a greatest common factor?
Yes. It first removes any shared numeric factor from all three coefficients, then it normalizes the sign when the leading coefficient remains negative.
7) What if my trinomial becomes a perfect square?
The calculator will show the repeated-root structure, such as -(2x + 1)², and the graph will touch the x-axis instead of crossing it.
8) Can I use the exports for homework review?
Yes. CSV exports the key numeric and symbolic results, while PDF captures a compact summary that works well for notes, practice sheets, and revision folders.