Enter the coefficients (a, b, c, d) for the cubic equation in the form ax³ + bx² + cx + d = 0 to find its roots.
Understanding Cubic Equations
A cubic equation is a polynomial equation of degree three, meaning the highest exponent of the variable is 3. It takes the general form: ax³ + bx² + cx + d = 0, where 'a', 'b', 'c', and 'd' are coefficients, and 'a' cannot be zero. If 'a' were zero, it would become a quadratic equation.
Cubic equations are fundamental in various fields, including:
Engineering: Designing structures, analyzing fluid dynamics, and electrical circuits.
Physics: Describing motion, oscillations, and quantum mechanics.
Mathematics: Advanced algebra, geometry, and calculus.
Economics: Modeling supply and demand curves, and optimization problems.
Types of Roots
A cubic equation always has three roots (solutions), which can be real or complex. The nature of these roots depends on the discriminant of the equation:
Three distinct real roots: All three solutions are real numbers and are different from each other.
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).
All real roots, with at least two being equal: This can mean two distinct real roots (one repeated, one unique) or one distinct real root (all three are equal).
How to Use the Cubic Equation Solver
To use this calculator, simply input the coefficients 'a', 'b', 'c', and 'd' from your cubic equation ax³ + bx² + cx + d = 0 into the respective fields. For example, if your equation is 2x³ - 3x² + 5x - 1 = 0, you would enter:
Coefficient 'a': 2
Coefficient 'b': -3
Coefficient 'c': 5
Coefficient 'd': -1
Click the "Calculate Roots" button, and the calculator will display the three roots of your equation.
Examples
Example 1: Three Distinct Real Roots
Consider the equation: x³ - 6x² + 11x - 6 = 0
a = 1
b = -6
c = 11
d = -6
Expected Roots: x = 1, x = 2, x = 3
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
Expected Roots: x = -1, x = i, x = -i
Example 3: All Real Roots, with Two Being Equal
Consider the equation: x³ - 3x² + 3x - 1 = 0
a = 1
b = -3
c = 3
d = -1
Expected Roots: x = 1 (triple root)
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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('result');
var roots = [];
var epsilon = 1e-9; // For floating point comparisons
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients.";
resultDiv.style.color = "#dc3545";
resultDiv.style.backgroundColor = "#f8d7da";
return;
}
// Handle special cases: a = 0 (quadratic), a = 0, b = 0 (linear), etc.
if (Math.abs(a) < epsilon) {
// It's a quadratic equation: bx^2 + cx + d = 0
if (Math.abs(b) < epsilon) {
// It's a linear equation: cx + d = 0
if (Math.abs(c) < epsilon) {
// It's a constant: d = 0
if (Math.abs(d) < epsilon) {
resultDiv.innerHTML = "Infinite solutions (0 = 0).";
} else {
resultDiv.innerHTML = "No solution (e.g., " + d + " = 0).";
}
} else {
// Linear equation: cx + d = 0 => x = -d/c
var x = -d / c;
roots.push(x.toFixed(6));
resultDiv.innerHTML = "This is a linear equation (a=0, b=0).Root: x = " + roots[0];
}
} else {
// Quadratic equation: bx^2 + cx + d = 0
var delta = c * c – 4 * b * d;
if (delta >= -epsilon) { // delta >= 0 (or very close to 0)
var x1 = (-c + Math.sqrt(delta)) / (2 * b);
var x2 = (-c – Math.sqrt(delta)) / (2 * b);
roots.push(x1.toFixed(6));
roots.push(x2.toFixed(6));
resultDiv.innerHTML = "This is a quadratic equation (a=0).Roots: x₁ = " + roots[0] + ", x₂ = " + roots[1];
} else { // delta < 0
var realPart = (-c / (2 * b)).toFixed(6);
var imagPart = (Math.sqrt(Math.abs(delta)) / (2 * b)).toFixed(6);
roots.push(realPart + " + " + imagPart + "i");
roots.push(realPart + " – " + imagPart + "i");
resultDiv.innerHTML = "This is a quadratic equation (a=0).Roots: x₁ = " + roots[0] + ", x₂ = " + roots[1];
}
}
resultDiv.style.color = "#0056b3";
resultDiv.style.backgroundColor = "#e9f7ef";
return;
}
// Normalize the cubic equation to x^3 + px^2 + qx + r = 0
var p = b / a;
var q = c / a;
var r = d / a;
// Depress the cubic: x = y – p/3
var P = q – (p * p / 3);
var Q = (2 * p * p * p / 27) – (p * q / 3) + r;
// Calculate the discriminant of the depressed cubic: D = -4P^3 – 27Q^2
var D = -4 * P * P * P – 27 * Q * Q;
var y1, y2, y3;
if (D > epsilon) { // Three distinct real roots (D > 0)
var phi = Math.acos((-Q / 2) / Math.sqrt(-P * P * P / 27));
var sqrt_term = 2 * Math.sqrt(-P / 3);
y1 = sqrt_term * Math.cos(phi / 3);
y2 = sqrt_term * Math.cos((phi + 2 * Math.PI) / 3);
y3 = sqrt_term * Math.cos((phi + 4 * Math.PI) / 3);
} else if (Math.abs(D) < epsilon) { // All real roots, at least two are equal (D = 0)
var cbrt_Q_half = Math.cbrt(-Q / 2);
y1 = 2 * cbrt_Q_half;
y2 = -cbrt_Q_half;
y3 = -cbrt_Q_half;
} else { // One real root and two complex conjugate roots (D < 0)
var sqrt_term = Math.sqrt(Q * Q / 4 + P * P * P / 27);
var u = Math.cbrt(-Q / 2 + sqrt_term);
var v = Math.cbrt(-Q / 2 – sqrt_term);
y1 = u + v;
var realPart_y23 = -0.5 * (u + v);
var imagPart_y23 = 0.5 * Math.sqrt(3) * (u – v);
// Transform back to x for the real root
roots.push((y1 – p / 3).toFixed(6));
// For complex roots, format them
var x2_real = (realPart_y23 – p / 3).toFixed(6);
var x2_imag = imagPart_y23.toFixed(6);
var x3_real = (realPart_y23 – p / 3).toFixed(6);
var x3_imag_abs = Math.abs(imagPart_y23).toFixed(6); // Get absolute value for display
roots.push(x2_real + " + " + x2_imag + "i");
roots.push(x3_real + " – " + x3_imag_abs + "i");
resultDiv.innerHTML = "Roots:x₁ = " + roots[0] + "x₂ = " + roots[1] + "x₃ = " + roots[2];
resultDiv.style.color = "#0056b3";
resultDiv.style.backgroundColor = "#e9f7ef";
return;
}
// Transform back to x for all roots
roots.push((y1 – p / 3).toFixed(6));
roots.push((y2 – p / 3).toFixed(6));
roots.push((y3 – p / 3).toFixed(6));
resultDiv.innerHTML = "Roots:x₁ = " + roots[0] + "x₂ = " + roots[1] + "x₃ = " + roots[2];
resultDiv.style.color = "#0056b3";
resultDiv.style.backgroundColor = "#e9f7ef";
}