Calculator Inputs
Example Data Table
| Case | Equation | Factor Form | Roots |
|---|---|---|---|
| Quadratic 1 | x² - 5x + 6 = 0 | (x - 2)(x - 3) | 2, 3 |
| Quadratic 2 | 2x² + 5x + 2 = 0 | (2x + 1)(x + 2) | -0.5, -2 |
| Quadratic 3 | x² + 4x + 4 = 0 | (x + 2)² | -2, -2 |
| Cubic 1 | x³ - 6x² + 11x - 6 = 0 | (x - 1)(x - 2)(x - 3) | 1, 2, 3 |
| Cubic 2 | x³ + x² - 4x - 4 = 0 | (x + 1)(x - 2)(x + 2) | -1, 2, -2 |
Formula Used
Quadratic rule: If ax² + bx + c = 0, then roots are x = (-b ± √(b² - 4ac)) / 2a.
Discriminant: Δ = b² - 4ac. It determines whether the roots are real, repeated, or complex.
Factor theorem: If f(r) = 0, then (x - r) is a factor.
Rational root test: Possible cubic rational roots are ± factors of d over factors of a.
Cardano approach: Cubics reduce to t³ + pt + q = 0, then Δ = (q/2)² + (p/3)³ guides the root pattern.
How to Use This Calculator
- Select whether you want to factor a quadratic or cubic equation.
- Enter the polynomial coefficients in standard form.
- Choose the variable symbol and the decimal precision.
- Press Factor Equation to place the result above the form.
- Review factor form, root type, special values, and proof notes.
- Download the table as CSV or PDF when needed.
FAQs
1. What equations does this calculator support?
It factors quadratic and cubic equations written in standard form. Quadratics use a, b, c. Cubics use a, b, c, d, with the expression set equal to zero.
2. Does it show repeated roots?
Yes. When a discriminant becomes zero, the result marks a repeated quadratic root. For cubics, repeated real roots are also identified when the cubic discriminant indicates duplication.
3. Can it handle complex roots?
Yes. If the equation cannot factor fully over real numbers, the calculator still reports complex roots and builds an approximate complex factor form.
4. Why do some factors look approximate?
Approximate factors appear when roots are irrational or complex, or when a cubic lacks a simple rational root. The calculator then uses numerical methods to complete factorization.
5. What is the common factor value?
The common factor is the greatest shared integer factor across coefficients when the inputs are integer based. It helps identify whether the expression contains an immediate simplification.
6. Are decimal coefficients allowed?
Yes. Decimal entries are accepted. Exact symbolic factoring is strongest with integer coefficients, but decimal inputs still produce roots, classifications, and practical approximate factors.
7. What does the cubic delta value mean?
It is the discriminant of the depressed cubic. Its sign helps determine whether the cubic has three real roots, a repeated root, or one real root with a complex pair.
8. Can I save the calculation results?
Yes. Use the CSV button to export the summary table for spreadsheets, or the PDF button to save a clean report for lessons, homework, or reference.