Math & Algebra

Free Polynomial Roots & Factoring Calculator — Real & Complex

Finding roots (zeros) of polynomial equations where $f(x) = 0$ is a core operation across algebra, calculus, physics, and engineering. The roots of a polynomial represent where the graph intersects or touches the x-axis. Our free polynomial roots calculator allows algebra students to find roots of cubic equation problems, solve quadratic cubic roots solver equations, and evaluate real and complex roots calculator matrices.

Polynomial Roots & Factoring Calculator

Solve quadratic and cubic equations $ax^3 + bx^2 + cx + d = 0$, find real and complex roots, and calculate discriminants.

Coeff a (x³) / (x²)

Coeff b (x²) / (x¹)

Coeff c (x¹) / (c)

Coeff d (Constant)Set d = 0 for Quadratic

Polynomial Roots Solution

x₁ = 3.000

x₂ = 2.000

2 Distinct Real Roots (Δ > 0)

Discriminant (Δ = b² - 4ac):1

Parabola Vertex & Algebraic Factoring Breakdown

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

h = -(-5) / (2 × 1) = 2.500

k = 1×(2.500)² + -5×(2.500) + 6 = -0.250

Vertex Coordinates = (2.50, -0.25)

Mathematical Formulas & Calculation Methods

Quadratic & Cubic Root Formulas

Quadratic formula and discriminant equations.

Quadratic: x = [ -b ± √(b² - 4ac) ] / (2a) | Discriminant Δ = b² - 4ac

Worked Step-by-Step Examples

Example 1: Solve x² - 5x + 6 = 0 (a = 1, b = -5, c = 6, d = 0)

"Calculate roots and discriminant Δ."

Steps:

Discriminant Δ = (-5)² - 4(1)(6) = 25 - 24 = 1 (> 0).
x₁ = [ 5 + √1 ] / 2 = 6 / 2 = 3.0.
x₂ = [ 5 - √1 ] / 2 = 4 / 2 = 2.0.

Result: x₁ = 3.0, x₂ = 2.0 (2 Distinct Real Roots).

What is Polynomial Roots & Quadratic/Cubic Factoring Calculator?

A polynomial roots calculator computes real and complex roots for quadratic ($ax^2+bx+c=0$) and cubic polynomials.

The **Discriminant ($Delta = b^2 - 4ac$)** indicates root types: $Delta > 0$ yields 2 real roots, $Delta = 0$ yields 1 repeated root, and $Delta < 0$ yields 2 complex conjugate roots.

How to Use Polynomial Roots & Quadratic/Cubic Factoring Calculator

1Enter polynomial coefficients a (x³ or x²), b (x² or x¹), c (x¹ or constant), and d (constant).
2Set d = 0 for quadratic equations.
3View instant real and complex roots ($x_1, x_2$), discriminant $Delta$, and parabola vertex coordinates.

Key Features

Supports Quadratic and Cubic polynomial equations.
Calculates Real and Complex conjugate roots ($a + bi$).
Displays Discriminant $Delta$ and Parabola Vertex (h, k).

Benefits of Using Polynomial Roots & Quadratic/Cubic Factoring Calculator

Saves algebra and pre-calculus students from tedious quadratic formula arithmetic.

Handles complex imaginary roots ($i$) automatically.

Practical Use Cases

Algebra & Pre-Calculus Students

Solve quadratic formula and cubic polynomial root factoring homework.

Physics & Engineering Students

Calculate roots of characteristic equations for differential equations.

Frequently Asked Questions

What does an imaginary root 'i' mean?

An imaginary root containing 'i' (where i = √-1) indicates that the parabola does not physically cross the real x-axis.

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