.calculator-container {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.calculator-container p {
color: #555;
margin-bottom: 15px;
line-height: 1.6;
}
.calc-input-group {
margin-bottom: 15px;
}
.calc-input-group label {
display: block;
margin-bottom: 5px;
color: #333;
font-weight: bold;
}
.calc-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
font-size: 16px;
}
.calc-button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
.calc-button:hover {
background-color: #0056b3;
}
.calc-result-area {
margin-top: 25px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 4px;
background-color: #e9ecef;
min-height: 50px;
color: #333;
}
.calc-result-area p {
margin: 5px 0;
font-size: 1.1em;
font-weight: bold;
}
.calc-result-area p:first-of-type {
margin-top: 0;
}
.calc-result-area p:last-of-type {
margin-bottom: 0;
}
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");
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients.";
return;
}
var outputHTML = "";
var EPSILON = 1e-9; // Small epsilon for floating point comparisons
// Handle edge case: a = 0 (becomes a quadratic equation or simpler)
if (Math.abs(a) < EPSILON) {
if (Math.abs(b) < EPSILON) {
// If a=0 and b=0, it's a linear equation: cx + d = 0
if (Math.abs(c) < EPSILON) {
if (Math.abs(d) x = -d/c
var x = -d / c;
outputHTML = "This is a linear equation (a=0, b=0). One real root:";
outputHTML += "x = " + x.toFixed(6) + "";
}
} else {
// If a=0, it's a quadratic equation: bx^2 + cx + d = 0
var delta = c * c – 4 * b * d;
outputHTML = "This is a quadratic equation (a=0). Roots are:";
if (delta >= -EPSILON) { // delta >= 0
var x1 = (-c + Math.sqrt(Math.max(0, delta))) / (2 * b);
var x2 = (-c – Math.sqrt(Math.max(0, delta))) / (2 * b);
outputHTML += "x1 = " + x1.toFixed(6) + "";
outputHTML += "x2 = " + x2.toFixed(6) + "";
} else { // delta = EPSILON) { // deltaCubic > 0
// One real root and two complex conjugate roots
var u = Math.cbrt(-q / 2 + Math.sqrt(deltaCubic));
var v = Math.cbrt(-q / 2 – Math.sqrt(deltaCubic));
var y1 = u + v;
var realPartComplex = (-y1 / 2);
var imagPartComplex = (Math.sqrt(3) / 2) * (u – v);
outputHTML = "One real root and two complex conjugate roots:";
outputHTML += "x1 = " + (y1 + offset).toFixed(6) + "";
outputHTML += "x2 = " + (realPartComplex + offset).toFixed(6) + " + " + imagPartComplex.toFixed(6) + "i";
outputHTML += "x3 = " + (realPartComplex + offset).toFixed(6) + " – " + imagPartComplex.toFixed(6) + "i";
} else if (deltaCubic > -EPSILON && deltaCubic < EPSILON) { // deltaCubic = 0
// All real roots, at least two equal
var u = Math.cbrt(-q / 2); // This is the value for u and v when deltaCubic is 0
var y1 = 2 * u;
var y2 = -u; // y2 and y3 are equal
outputHTML = "Three real roots (at least two are equal):";
outputHTML += "x1 = " + (y1 + offset).toFixed(6) + "";
outputHTML += "x2 = " + (y2 + offset).toFixed(6) + "";
outputHTML += "x3 = " + (y2 + offset).toFixed(6) + "";
} else { // deltaCubic < 0 (Three distinct real roots – casus irreducibilis)
// Ensure p is negative for this case, which it must be if deltaCubic = -EPSILON) { // Should ideally not happen, but a safeguard
resultDiv.innerHTML = "Error: Unexpected condition for three real roots (p should be negative).";
return;
}
// Calculate theta using the form cos(theta) = -q / (2 * sqrt(-p^3/27))
var sqrt_neg_p_cubed_over_27 = Math.sqrt(Math.abs(Math.pow(p, 3) / 27));
var termForAcos = -q / (2 * sqrt_neg_p_cubed_over_27);
// Clamp termForAcos to [-1, 1] due to floating point inaccuracies
termForAcos = Math.max(-1, Math.min(1, termForAcos));
var theta = Math.acos(termForAcos);
var r_val = Math.sqrt(-p / 3); // This is sqrt(-p/3)
var y1 = 2 * r_val * Math.cos(theta / 3);
var y2 = 2 * r_val * Math.cos((theta + 2 * Math.PI) / 3);
var y3 = 2 * r_val * Math.cos((theta + 4 * Math.PI) / 3);
outputHTML = "Three distinct real roots:";
outputHTML += "x1 = " + (y1 + offset).toFixed(6) + "";
outputHTML += "x2 = " + (y2 + offset).toFixed(6) + "";
outputHTML += "x3 = " + (y3 + offset).toFixed(6) + "";
}
resultDiv.innerHTML = outputHTML;
}
Understanding the Cubic Formula and its Calculator
A cubic equation is a polynomial equation of the third degree, meaning the highest power of the variable is three. It takes the general form:
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 Challenge of Solving Cubic Equations
Unlike quadratic equations, which can be solved relatively easily using the quadratic formula, cubic equations are significantly more complex. The general algebraic solution, often attributed to Cardano and Tartaglia, involves a series of transformations and calculations that can lead to real or complex roots.
The nature of the roots (how many are real and how many are complex) depends on a discriminant value derived from the coefficients. This discriminant helps categorize the solutions into three main cases:
- One Real Root and Two Complex Conjugate Roots: This occurs when the discriminant is positive.
- Three Real Roots (at least two are equal): This happens when the discriminant is zero.
- Three Distinct Real Roots (Casus Irreducibilis): This is the most intricate case, occurring when the discriminant is negative. It requires trigonometric functions to find the real roots, even though the intermediate steps might involve complex numbers.
How This Calculator Works
This Cubic Formula Calculator simplifies the process by implementing the algebraic solution (Cardano's method) behind the scenes. Here's a simplified overview of the steps it performs:
- Input Validation: It first checks if all entered coefficients (a, b, c, d) are valid numbers.
- Edge Cases (a=0): If the coefficient 'a' is zero, the equation is no longer cubic. The calculator intelligently identifies this and solves it as either a quadratic equation (if b ≠ 0) or a linear equation (if a=0, b=0, c ≠ 0). It also handles cases where the equation becomes an identity (0=0) or a contradiction (e.g., 5=0).
- Normalization: For true cubic equations (a ≠ 0), it normalizes the equation by dividing all terms by 'a', transforming it into the form
x³ + Bx² + Cx + D = 0.
- Depressed Cubic Transformation: It then transforms the normalized cubic into a "depressed cubic" form (
y³ + py + q = 0) by substituting x = y - B/3. This simplifies the equation by eliminating the y² term.
- Discriminant Calculation: The calculator computes a discriminant based on
p and q. This value determines the nature of the roots.
- Root Calculation: Based on the discriminant, it applies the appropriate part of Cardano's formula to find the roots (y1, y2, y3) of the depressed cubic.
- Back-Substitution: Finally, it converts the roots of the depressed cubic back to the original variable
x using the relation x = y - B/3.
- Result Display: The calculated roots are displayed, clearly indicating if they are real or complex numbers.
Examples of Cubic Equations and Their Roots:
Let's look at some examples to illustrate the different types of roots:
Example 1: Three Distinct Real Roots
Equation: x³ - 6x² + 11x - 6 = 0
- Coefficient a: 1
- Coefficient b: -6
- Coefficient c: 11
- Coefficient d: -6
Roots: x1 = 1, x2 = 2, x3 = 3
Example 2: One Real Root and Two Complex Conjugate Roots
Equation: x³ + x² + x + 1 = 0
- Coefficient a: 1
- Coefficient b: 1
- Coefficient c: 1
- Coefficient d: 1
Roots: x1 = -1, x2 = 0 + 1i, x3 = 0 – 1i
Example 3: Three Real Roots (at least two equal)
Equation: x³ - 3x² + 3x - 1 = 0
- Coefficient a: 1
- Coefficient b: -3
- Coefficient c: 3
- Coefficient d: -1
Roots: x1 = 1, x2 = 1, x3 = 1 (a triple root)
Example 4: Quadratic Equation (when a=0)
Equation: 0x³ + 2x² + 3x - 5 = 0 (effectively 2x² + 3x - 5 = 0)
- Coefficient a: 0
- Coefficient b: 2
- Coefficient c: 3
- Coefficient d: -5
Roots: x1 = 1, x2 = -2.5
This calculator is a powerful tool for students, engineers, and mathematicians to quickly and accurately find the roots of any cubic equation, regardless of the complexity of its coefficients or the nature of its solutions.