Polynomial Equation Solver & Roots Calculator

Algebra & Polynomial Theory: Roots & The Fundamental Theorem of Algebra

• The Fundamental Theorem of Algebra: A polynomial of degree n has exactly n roots in the complex number system (counting multiplicity).

• The Discriminant):

• Δ > 0: Parabola crosses the x-axis twice (two distinct real roots).

• \: Parabola is tangent to the x-axis (one repeated real root).

• Δ < 0: Parabola does not intersect the x-axis (two complex conjugate roots u ± vi).

How to Use This Calculator

Enter your Polynomial Degree Order — choose from Degree 2: Quadratic Polynomial (ax² + bx + c = 0), Degree 3: Cubic Polynomial (ax³ + bx² + cx + d = 0), Leading Coefficient a (x² or x³) (e.g. 1), Coefficient b (x or x²) (e.g. -5), Coefficient c (Constant or x) (e.g. 6), Constant Term d (for Cubic) (e.g. 0 (used for cubic)), and Evaluation Point (x₀ to compute P(x₀)) (e.g. 2). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Polynomial \& Quadratic equations:

Quadratic Formula: x = (-b ± √(Δ)) / (2a) (a ≠ 0)

Discriminant: Δ = b^2 - 4ac

Δ > 0 2 Real Roots, Δ = 0 1 Repeated Root, Δ < 0 2 Complex Roots

Parabola Vertex: h = -(b) / (2a), k = P(h)

First Derivative: P'(x) = 2ax + b

Worked Example

For example, solving the quadratic equation : Discriminant is \^2 - 4(1)(6) = 25 - 24 =

1.

• Roots are \ and \. Parabola vertex is at (2.50, -0.25) (minimum point), with derivative P'(x) = 2x -

5.

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.