Cubic Equation Calculator

Cubic Equation Solver

Enter the coefficients for your cubic equation in the form ax³ + bx² + cx + d = 0 to find its roots.

function solveCubicEquation() { var a = parseFloat(document.getElementById("coeffA").value); var b = parseFloat(document.getElementById("coeffB").value); var c = parseFloat(document.getElementById("coeffC").value); var d = parseFloat(document.getElementById("coeffD").value); var resultDiv = document.getElementById("cubicResult"); resultDiv.innerHTML = ""; // Clear previous results if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { resultDiv.innerHTML = "Coefficient 'a' cannot be zero for a cubic equation. This would be a quadratic or linear equation."; return; } // Normalize the equation to x^3 + Ax^2 + Bx + C = 0 var A = b / a; var B = c / a; var C = d / a; // Transform to depressed cubic form: y^3 + py + q = 0 // x = y – A/3 var p = B – (A * A) / 3; var q = C – (A * B) / 3 + (2 * A * A * A) / 27; var delta = (q / 2) * (q / 2) + (p / 3) * (p / 3) * (p / 3); // Discriminant var roots = []; var offset = A / 3; // x = y – offset // Helper function for cube root that handles negative numbers correctly function cubeRoot(x) { return x = -EPSILON) { // delta is effectively >= 0 var u = cubeRoot(-q / 2 + Math.sqrt(Math.max(0, delta))); // Ensure sqrt argument is non-negative var v = cubeRoot(-q / 2 – Math.sqrt(Math.max(0, delta))); var y1 = u + v; roots.push((y1 – offset).toFixed(6)); // Check if the imaginary part is effectively zero for the other two roots var realPart_y23 = -(u + v) / 2; var imagPart_y23 = (Math.sqrt(3) / 2) * (u – v); if (Math.abs(imagPart_y23) < EPSILON) { // If imaginary part is effectively zero, means all roots are real roots.push((realPart_y23 – offset).toFixed(6)); roots.push((realPart_y23 – offset).toFixed(6)); } else { // One real, two complex conjugates roots.push((realPart_y23 – offset).toFixed(6) + " + " + imagPart_y23.toFixed(6) + "i"); roots.push((realPart_y23 – offset).toFixed(6) + " – " + imagPart_y23.toFixed(6) + "i"); } } else { // delta 1) arg_acos = 1; if (arg_acos < -1) arg_acos = -1; var phi = Math.acos(arg_acos); var r = 2 * Math.sqrt(-p / 3); var y1 = r * Math.cos(phi / 3); var y2 = r * Math.cos((phi + 2 * Math.PI) / 3); var y3 = r * Math.cos((phi + 4 * Math.PI) / 3); roots.push((y1 – offset).toFixed(6)); roots.push((y2 – offset).toFixed(6)); roots.push((y3 – offset).toFixed(6)); } var output = "

Roots of the equation:

"; for (var i = 0; i < roots.length; i++) { output += "x" + (i + 1) + " = " + roots[i] + ""; } resultDiv.innerHTML = output; }

Understanding Cubic Equations and Their Solutions

A cubic equation is a polynomial equation of the third degree, meaning it contains a term where the variable is raised to the power of three. Its general form is expressed as:

ax³ + bx² + cx + d = 0

Where 'a', 'b', 'c', and 'd' are coefficients, and 'a' cannot be zero (otherwise, it would be a quadratic or linear equation). The variable 'x' represents the unknown value(s) we aim to find, known as the roots or solutions of the equation.

Why are Cubic Equations Important?

Cubic equations appear in various fields of science, engineering, and mathematics. They are fundamental in:

  • Physics: Describing motion, fluid dynamics, and wave propagation.
  • Engineering: Designing structures, analyzing electrical circuits, and optimizing processes.
  • Geometry: Calculating volumes of complex shapes or finding intersection points of curves.
  • Economics: Modeling supply and demand curves or growth patterns.
  • Computer Graphics: Used in Bezier curves and other spline functions for smooth shapes.

Understanding how to solve these equations is crucial for many analytical and design tasks.

Solving Cubic Equations: A Brief Overview

Unlike quadratic equations, which can be solved with the relatively straightforward quadratic formula, cubic equations require more complex methods. The most famous analytical solution method is known as Cardano's formula, developed in the 16th century. This formula can be quite intricate and involves several steps:

  1. Normalization: The equation is first divided by 'a' to simplify it to the form x³ + Ax² + Bx + C = 0.
  2. Depressed Cubic Form: A substitution (e.g., x = y - A/3) is made to eliminate the x² term, resulting in a "depressed" cubic equation of the form y³ + py + q = 0.
  3. Cardano's Formula Application: Specific formulas involving cube roots and square roots of the coefficients 'p' and 'q' are then applied to find the values of 'y'.
  4. Back-Substitution: Finally, the values of 'y' are substituted back into the original substitution to find the roots 'x'.

A cubic equation will always have three roots. These roots can be:

  • Three distinct real roots.
  • One real root and two complex conjugate roots.
  • Three real roots, where at least two of them are identical (e.g., one root with multiplicity two, or one root with multiplicity three).

How to Use This Calculator

Our Cubic Equation Solver simplifies this complex process for you. Simply input the coefficients 'a', 'b', 'c', and 'd' from your equation ax³ + bx² + cx + d = 0 into the respective fields. For example, if your equation is x³ - 6x² + 11x - 6 = 0, you would enter:

  • Coefficient 'a': 1
  • Coefficient 'b': -6
  • Coefficient 'c': 11
  • Coefficient 'd': -6

Click the "Calculate Roots" button, and the calculator will instantly display all three roots of your cubic equation, whether they are real or complex numbers.

Example Calculation:

Let's use the example equation: x³ - 6x² + 11x - 6 = 0

Inputting the coefficients:

  • a = 1
  • b = -6
  • c = 11
  • d = -6

The calculator will output the roots as:

x₁ = 1.000000
x₂ = 2.000000
x₃ = 3.000000

These are the three distinct real roots for this specific cubic equation.

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