Interactive Surface Graph
This graph shows z = ax² + bxy + cy² over x and y.
Calculator
Enter coefficients for ax² + bxy + cy². The tool extracts a numeric GCF, tests integer factors, and shows real factors when useful.
Formula Used
A positive discriminant gives real linear factors. A zero discriminant gives a repeated factor. A negative discriminant blocks real linear factoring.
How to Use This Calculator
- Enter the coefficient of x².
- Enter the coefficient of xy.
- Enter the coefficient of y².
- Click Factor Polynomial.
- Read the integer factorization first.
- Check the reduced form and discriminant next.
- Review the real factorization if needed.
- Use CSV or PDF export for records.
Example Data Table
| Polynomial | Factorization | Case |
|---|---|---|
| x2 + 5xy + 6y2 | (x + 2y)(x + 3y) | Integer factorization |
| 4x2 - 9y2 | (2x - 3y)(2x + 3y) | Difference of squares |
| 6x2 + 15xy + 6y2 | 3(2x + y)(x + 2y) | Numeric GCF plus factoring |
| 3xy + 6y2 | 3y(x + 2y) | Common factor y |
| 3x2 + 12xy + 12y2 | 3(x + 2y)(x + 2y) | Perfect square trinomial |
| x2 + xy + y2 | Irreducible over integers | Positive non-square discriminant |
FAQs
1. What form does this calculator factor?
It factors expressions shaped like ax² + bxy + cy². It also extracts any numeric GCF and handles simple common-variable cases automatically.
2. Does it factor every two-variable expression?
No. Some bivariate expressions need broader symbolic algebra methods. This tool focuses on the most common quadratic two-variable factoring pattern.
3. Why is the discriminant shown?
The discriminant reveals whether real linear factors exist. Negative values block real factors, zero creates a repeated factor, and positive values allow real factors.
4. What does the numeric GCF do?
The numeric GCF removes a shared coefficient first. That often makes the remaining quadratic easier to factor into two binomials.
5. Why might integer factoring fail?
Integer factoring fails when no integer values m, p, n, and q satisfy the matching product and middle-term conditions together.
6. What does “real factorization” mean here?
It shows approximate real linear factors when integer binomials do not exist. This helps explain the structure even when neat integer factors fail.
7. Can I use negative coefficients?
Yes. The calculator accepts positive, negative, and zero coefficients. It normalizes signs when needed to present cleaner factoring steps.
8. What are CSV and PDF exports for?
They save your polynomial, reduced form, discriminant, classification, and factorization summary. This is helpful for homework notes, reports, and quick sharing.