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;
}
if (a === 0) {
resultDiv.innerHTML = 'Coefficient \'a\' cannot be zero for a cubic equation. This would be a quadratic equation.';
return;
}
// Normalize coefficients for depressed cubic form y^3 + py + q = 0
// x = y – b/(3a)
var p = (3 * a * c – b * b) / (3 * a * a);
var q = (2 * b * b * b – 9 * a * b * c + 27 * a * a * d) / (27 * a * a * a);
// Discriminant D = (q/2)^2 + (p/3)^3
var D = (q / 2) * (q / 2) + (p / 3) * (p / 3) * (p / 3);
var roots = [];
var tolerance = 1e-9; // For floating point comparisons
if (D > tolerance) { // One real root, two complex conjugate roots
var u = Math.cbrt(-q / 2 + Math.sqrt(D));
var v = Math.cbrt(-q / 2 – Math.sqrt(D));
var y1 = u + v;
var x1 = y1 – b / (3 * a);
roots.push(x1.toFixed(6));
var y2_real = -(u + v) / 2;
var y2_imag = Math.sqrt(3) / 2 * (u – v);
var x2_real = y2_real – b / (3 * a);
var x2_imag = y2_imag;
roots.push(x2_real.toFixed(6) + " + " + x2_imag.toFixed(6) + "i");
var x3_real = y2_real – b / (3 * a);
var x3_imag = -y2_imag;
roots.push(x3_real.toFixed(6) + " " + (x3_imag < 0 ? "-" : "+") + " " + Math.abs(x3_imag).toFixed(6) + "i");
} else if (D < -tolerance) { // Three distinct real roots (casus irreducibilis)
var r_val = Math.sqrt(-Math.pow(p / 3, 3)); // This is sqrt(-p^3/27)
var theta_val = Math.acos(-q / (2 * r_val));
var sqrt_neg_p_over_3 = Math.sqrt(-p / 3);
var y1 = 2 * sqrt_neg_p_over_3 * Math.cos(theta_val / 3);
var y2 = 2 * sqrt_neg_p_over_3 * Math.cos((theta_val + 2 * Math.PI) / 3);
var y3 = 2 * sqrt_neg_p_over_3 * Math.cos((theta_val + 4 * Math.PI) / 3);
roots.push((y1 – b / (3 * a)).toFixed(6));
roots.push((y2 – b / (3 * a)).toFixed(6));
roots.push((y3 – b / (3 * a)).toFixed(6));
} else { // D is approximately 0 (All roots are real, at least two are equal)
var u = Math.cbrt(-q / 2);
var y1 = 2 * u;
var y2 = -u; // This is a double root
var y3 = -u;
roots.push((y1 – b / (3 * a)).toFixed(6));
roots.push((y2 – b / (3 * a)).toFixed(6));
roots.push((y3 – b / (3 * a)).toFixed(6));
}
var resultHtml = '
';
resultDiv.innerHTML = resultHtml;
}
.cubic-equation-calculator {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: Arial, sans-serif;
}
.cubic-equation-calculator h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.cubic-equation-calculator p {
margin-bottom: 15px;
line-height: 1.6;
}
.calculator-input-group {
margin-bottom: 15px;
}
.calculator-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.calculator-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
}
.cubic-equation-calculator button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 16px;
cursor: pointer;
transition: background-color 0.3s ease;
}
.cubic-equation-calculator button:hover {
background-color: #0056b3;
}
.calculator-result {
margin-top: 20px;
padding: 15px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 4px;
color: #155724;
}
.calculator-result h3 {
color: #155724;
margin-top: 0;
margin-bottom: 10px;
}
.calculator-result ul {
list-style-type: none;
padding: 0;
margin: 0;
}
.calculator-result li {
margin-bottom: 8px;
font-size: 1.1em;
}
.calculator-result .error {
color: #721c24;
background-color: #f8d7da;
border-color: #f5c6cb;
padding: 10px;
border-radius: 4px;
}
Understanding the Cubic Equation Formula
A cubic equation is a polynomial equation of degree three, meaning the highest power of the variable is three. Its standard form is expressed as:
ax³ + bx² + cx + d = 0
Where:
a, b, c, and d are coefficients (real numbers).x is the variable.- The coefficient
a cannot be zero, otherwise, it would become a quadratic equation.
The Nature of Cubic Roots
Unlike quadratic equations, which always have two roots (real or complex), a cubic equation always has exactly three roots. These roots can be:
- Three distinct real roots: All three solutions for
x are unique real numbers.
- 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.,
m + ni and m - ni).
- Three real roots, with at least two being equal: This includes cases where there is one distinct real root and a double real root, or even a triple real root.
The nature of these roots is determined by the discriminant of the cubic equation, similar to how the discriminant of a quadratic equation determines the nature of its roots.
Solving Cubic Equations: Cardano's Formula
While quadratic equations can be solved with the relatively straightforward quadratic formula, cubic equations require a more complex approach, often attributed to mathematicians like Scipione del Ferro, Niccolò Fontana Tartaglia, and Gerolamo Cardano in the 16th century. Cardano's formula provides an analytical solution for the roots of a cubic equation.
The general method involves transforming the original cubic equation into a "depressed cubic" form (y³ + py + q = 0) by substituting x = y - b/(3a). The coefficients p and q are derived from a, b, c, d. Once the roots y are found for the depressed cubic, the original roots x can be easily calculated.
The complexity arises from the fact that even when all three roots are real, Cardano's formula might involve intermediate calculations with complex numbers (known as the "casus irreducibilis"). Modern numerical methods or trigonometric solutions are often used to simplify finding these real roots.
Practical Applications
Cubic equations appear in various fields of science and engineering, including:
- Physics: Describing wave phenomena, fluid dynamics, and orbital mechanics.
- Engineering: Structural analysis, control systems, and electrical circuit design.
- Mathematics: Geometry (finding intersections of curves), optimization problems, and cryptography.
- Economics: Modeling supply and demand curves or growth rates.
Example Calculation
Let's consider the cubic equation: x³ - 6x² + 11x - 6 = 0
Here, the coefficients are: a = 1, b = -6, c = 11, d = -6.
Using the calculator above with these values, you would find the roots to be:
- x₁ = 1.000000
- x₂ = 2.000000
- x₃ = 3.000000
Another example: x³ - 1 = 0
Coefficients: a = 1, b = 0, c = 0, d = -1.
The roots would be:
- x₁ = 1.000000
- x₂ = -0.500000 + 0.866025i
- x₃ = -0.500000 – 0.866025i
These examples demonstrate how the calculator can quickly provide the solutions, whether they are real or complex.