Factor by Decomposition Calculator

Calculator inputs

Enter integer coefficients for the trinomial ax² + bx + c. Results appear above this form after submission.

Example data table

Expression	a × c	Split numbers	Factor form
2x² + 7x + 3	6	6 and 1	(2x + 1)(x + 3)
3x² - x - 2	-6	2 and -3	(3x + 2)(x - 1)
x² + 10x + 25	25	5 and 5	(x + 5)(x + 5)

Formula used

For a quadratic trinomial ax² + bx + c, decomposition looks for two integers m and n such that:

m + n = b
m × n = a × c

After finding the pair, rewrite the middle term and factor by grouping:

ax² + bx + c = ax² + mx + nx + c

= (ax² + mx) + (nx + c)

= common factor from each group

= final product of two linear factors

The calculator also uses the discriminant b² - 4ac to report roots and graph behavior.

How to use this calculator

Enter the integer values of a, b, and c for your quadratic trinomial.
Click Factor Now to run the decomposition method.
Read the factor form, split numbers, discriminant, roots, and vertex.
Follow the step list to see how the middle term was decomposed.
Use the CSV or PDF buttons to save the final result and summary table.
Inspect the Plotly graph to connect the algebraic factor form with the parabola.

Frequently asked questions

1. What does factor by decomposition mean?

It means splitting the middle term of a quadratic into two parts. Those parts must add to b and multiply to a×c. Then you factor by grouping.

2. Does this calculator accept decimals?

No. This version is designed for whole-number coefficients because standard decomposition in school algebra usually uses integer pairs and integer grouping steps.

3. What if no split numbers are found?

Then the trinomial is not factorable by integer decomposition. The calculator still reports the discriminant, roots when real, and a graph of the parabola.

4. Why does the calculator show a greatest common factor first?

Factoring out the greatest common factor simplifies the trinomial. That makes the decomposition step cleaner and produces a more complete final factorization.

5. Can it handle perfect-square trinomials?

Yes. If the trinomial is a perfect square, the decomposition pair leads to identical linear factors, such as (x + 5)(x + 5).

6. Why are the roots included?

Roots show where the parabola crosses the x-axis. They help verify the factors because each linear factor becomes zero at one root.

7. What does the graph add to the result?

The graph shows the parabola, vertex, and real x-intercepts when they exist. It helps you connect symbolic factoring with the shape of the function.

8. Can I use the exported file in class notes?

Yes. The CSV is useful for tabular records, and the PDF works well for study notes, homework review, or printed examples.