Cubic Formula Calculator

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Cubic Formula Calculator

Solve for the roots of a cubic equation in the form: ax³ + bx² + cx + d = 0

Results will appear here.

function calculateCubicRoots() { 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; } var roots = []; var outputHtml = ""; // Handle special cases: a = 0 (quadratic, linear, or trivial) if (a === 0) { if (b === 0) { // Linear equation: cx + d = 0 if (c === 0) { if (d === 0) { outputHtml = "This is a trivial equation (0 = 0). Any real number is a solution."; } else { outputHtml = "This is an impossible equation (" + d + " = 0). No solution."; } } else { // cx + d = 0 => x = -d/c roots.push(-d / c); outputHtml = "This is a linear equation. One real root:x = " + roots[0].toFixed(6) + ""; } } else { // Quadratic equation: bx^2 + cx + d = 0 var discriminant = c * c – 4 * b * d; if (discriminant >= 0) { var x1 = (-c + Math.sqrt(discriminant)) / (2 * b); var x2 = (-c – Math.sqrt(discriminant)) / (2 * b); roots.push(x1); if (discriminant > 0) { roots.push(x2); } outputHtml = "This is a quadratic equation. Real roots:"; for (var i = 0; i = -EPSILON) { // delta >= 0 (or very close to zero) // One real root, two complex conjugate roots OR three real roots with some equal var u = Math.cbrt(-q / 2 + Math.sqrt(Math.max(0, delta))); // Ensure sqrt argument is non-negative var v = Math.cbrt(-q / 2 – Math.sqrt(Math.max(0, delta))); // Ensure sqrt argument is non-negative var y1 = u + v; roots.push(y1 – B / 3); if (Math.abs(delta) < EPSILON) { // delta is effectively zero // Three real roots, at least two are equal var y2 = -y1 / 2; roots.push(y2 – B / 3); roots.push(y2 – B / 3); // The third root is also y2 outputHtml = "Three real roots (at least two are equal):"; } else { // One real root, two complex conjugate roots // The complex roots are: // x2 = (- (u + v)/2 + i * Math.sqrt(3) * (u – v)/2) – B/3 // x3 = (- (u + v)/2 – i * Math.sqrt(3) * (u – v)/2) – B/3 var realPartComplex = (- (u + v) / 2 – B / 3).toFixed(6); var imagPartComplex = (Math.sqrt(3) * (u – v) / 2).toFixed(6); outputHtml = "One real root and two complex conjugate roots:"; outputHtml += "x1 = " + roots[0].toFixed(6) + ""; outputHtml += "x2 = " + realPartComplex + " + " + imagPartComplex + "i"; outputHtml += "x3 = " + realPartComplex + " – " + imagPartComplex + "i"; resultDiv.innerHTML = outputHtml; return; // Exit early as complex roots are handled } } else { // delta < 0 (three distinct real roots) // Three distinct real roots (trigonometric solution) var phi = Math.acos(-q / 2 / Math.sqrt(-(p / 3) * (p / 3) * (p / 3))); 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 – B / 3); roots.push(y2 – B / 3); roots.push(y3 – B / 3); outputHtml = "Three distinct real roots:"; } // Display real roots for (var i = 0; i < roots.length; i++) { outputHtml += "x" + (i + 1) + " = " + roots[i].toFixed(6) + ""; } resultDiv.innerHTML = outputHtml; }

Understanding the Cubic Formula and Equations

A cubic equation is a polynomial equation of the third degree, meaning the highest power of the variable (usually 'x') is 3. Its general form is:

ax³ + bx² + cx + d = 0

Where:

  • a, b, c, and d are coefficients (real numbers).
  • a cannot be zero, otherwise, it would be a quadratic equation (or lower degree).
  • x represents the unknown variable for which we are solving.

Why Solve Cubic Equations?

Cubic equations appear in various fields of science, engineering, and mathematics. For instance:

  • Physics: Describing motion, fluid dynamics, and wave propagation.
  • Engineering: Structural analysis, control systems, and electrical circuit design.
  • Economics: Modeling supply and demand curves, and optimization problems.
  • Geometry: Finding intersections of curves and surfaces.

While quadratic equations have a relatively simple formula, the cubic formula is significantly more complex, often attributed to mathematicians like Scipione del Ferro, Niccolò Fontana Tartaglia, and Gerolamo Cardano in the 16th century.

Nature of Cubic Roots

A fundamental theorem of algebra states that a polynomial equation of degree 'n' has exactly 'n' roots (counting multiplicity) in the complex number system. For a cubic equation (degree 3), this means there are always three roots.

These roots can be:

  1. Three distinct real roots: All three solutions are unique real numbers.
  2. One real root and two complex conjugate roots: One solution is a real number, and the other two are complex numbers that are conjugates of each other (e.g., p + qi and p - qi).
  3. Three real roots, with at least two being equal: This occurs when the graph of the cubic function touches the x-axis at one point and crosses it at another, or touches it at one point (a triple root).

Our calculator will identify and display these roots, distinguishing between real and complex solutions.

How to Use the Cubic Formula Calculator

Using this calculator is straightforward:

  1. Identify Coefficients: Look at your cubic equation and determine the values for a, b, c, and d. Remember that if a term is missing, its coefficient is 0 (e.g., if there's no x² term, b=0).
  2. Enter Values: Input the numerical values for a, b, c, and d into the respective fields in the calculator.
  3. Click "Calculate Roots": Press the button to instantly get the solutions.
  4. Review Results: The calculator will display the real roots. If complex roots exist, it will show them in the form p ± qi.

Examples of Cubic Equations

Let's look at some examples to illustrate how the calculator works:

Example 1: Three Distinct Real Roots

Consider the equation: x³ - 6x² + 11x - 6 = 0

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

Result: The calculator will output three distinct real roots: x1 = 1.000000, x2 = 2.000000, x3 = 3.000000.

Example 2: One Real Root and Two Complex Conjugate Roots

Consider the equation: x³ + x² + x + 1 = 0

  • a = 1
  • b = 1
  • c = 1
  • d = 1

Result: The calculator will output one real root and two complex conjugate roots: x1 = -1.000000, x2 = 0.000000 + 1.000000i, x3 = 0.000000 - 1.000000i.

Example 3: Three Real Roots (with some equal)

Consider the equation: x³ - 3x² + 3x - 1 = 0 (which is (x-1)³ = 0)

  • a = 1
  • b = -3
  • c = 3
  • d = -1

Result: The calculator will output three real roots, all equal: x1 = 1.000000, x2 = 1.000000, x3 = 1.000000.

Example 4: Quadratic Equation (when a=0)

Consider the equation: 0x³ + 1x² - 5x + 6 = 0 (effectively x² - 5x + 6 = 0)

  • a = 0
  • b = 1
  • c = -5
  • d = 6

Result: The calculator will identify it as a quadratic equation and provide its real roots: x1 = 3.000000, x2 = 2.000000.

Precision and Floating Point Arithmetic

It's important to note that calculations involving floating-point numbers in computers can sometimes lead to tiny inaccuracies. The calculator uses a small epsilon value (1e-9) to handle cases where a value is extremely close to zero, treating it as zero to provide more intuitive results (e.g., distinguishing between a discriminant that is -1e-15 and truly negative).

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