Quadratic Trinomial Calculator & Factorizer
Analyze quadratic trinomials (ax² + bx + c). Calculates roots, vertex, discriminant, and provides factorization steps. Precision support for large coefficients.
About
Quadratic trinomials represent parabolic functions used in physics, engineering, and economics. This calculator determines the roots of the expression ax2 + bx + c using the discriminant method. It factors the expression into linear binomials over the complex field if necessary. Accuracy is maintained up to 12 decimal places to minimize floating-point errors common in manual calculation. The tool identifies key features such as the vertex coordinates, which represent the minimum or maximum point of the parabola. This is essential for optimization problems.
Formulas
The properties are derived using the standard coefficients a, b, and c:
Vertex: h = −b2a, k = c − b24a
Discriminant: Δ = b2 − 4ac
Roots: x = −b ± √Δ2a
Reference Data
| Discriminant (Δ) | Roots Nature | Graph Interaction | Example |
|---|---|---|---|
| Δ > 0 | 2 Distinct Real Roots | Cuts x-axis twice | x2 − 3x + 2 |
| Δ = 0 | 1 Repeated Real Root | Touches x-axis once | x2 − 2x + 1 |
| Δ < 0 | 2 Complex Conjugate Roots | Does not touch x-axis | x2 + 1 |
| Perfect Square | Rational Roots | Symmetric about integer | 4x2 + 12x + 9 |
| Pure | Sum/Diff of Squares | Symmetric about y-axis | x2 − 9 |