Enter the coefficients for your quadratic equation in the form ax² + bx + c = 0 to find its roots.
Results:
function calculateQuadraticRoots() {
var a = parseFloat(document.getElementById("coefficientA").value);
var b = parseFloat(document.getElementById("coefficientB").value);
var c = parseFloat(document.getElementById("coefficientC").value);
var resultDiv = document.getElementById("quadraticResult");
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients.";
return;
}
if (a === 0) {
if (b === 0) {
if (c === 0) {
resultDiv.innerHTML = "This is a trivial equation (0=0). Any real number is a solution.";
} else {
resultDiv.innerHTML = "This is a contradiction (e.g., 5=0). No solution exists.";
}
} else {
// Linear equation: bx + c = 0 => x = -c/b
var x = -c / b;
resultDiv.innerHTML = "This is a linear equation (a=0). The root is: x = " + x.toFixed(4) + "";
}
return;
}
var discriminant = b * b – 4 * a * c;
var root1, root2;
if (discriminant > 0) {
root1 = (-b + Math.sqrt(discriminant)) / (2 * a);
root2 = (-b – Math.sqrt(discriminant)) / (2 * a);
resultDiv.innerHTML = "The equation has two distinct real roots:" +
"Root 1 (x₁): " + root1.toFixed(4) + "" +
"Root 2 (x₂): " + root2.toFixed(4) + "";
} else if (discriminant === 0) {
root1 = -b / (2 * a);
resultDiv.innerHTML = "The equation has one real root (or two identical real roots):" +
"Root (x): " + root1.toFixed(4) + "";
} else {
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
resultDiv.innerHTML = "The equation has two complex conjugate roots:" +
"Root 1 (x₁): " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i" +
"Root 2 (x₂): " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i";
}
}
.calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 25px;
max-width: 600px;
margin: 30px auto;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08);
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
font-size: 26px;
}
.calculator-container .calculator-content p {
color: #555;
line-height: 1.6;
margin-bottom: 15px;
}
.calculator-container .form-group {
margin-bottom: 18px;
display: flex;
flex-direction: column;
}
.calculator-container label {
font-weight: bold;
margin-bottom: 8px;
color: #444;
font-size: 15px;
}
.calculator-container input[type="number"] {
width: calc(100% – 20px);
padding: 12px;
border: 1px solid #ccc;
border-radius: 5px;
font-size: 16px;
box-sizing: border-box;
transition: border-color 0.3s ease;
}
.calculator-container input[type="number"]:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25);
}
.calculator-container .calculate-button {
background-color: #007bff;
color: white;
padding: 13px 25px;
border: none;
border-radius: 5px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease, transform 0.2s ease;
width: 100%;
box-sizing: border-box;
margin-top: 10px;
}
.calculator-container .calculate-button:hover {
background-color: #0056b3;
transform: translateY(-1px);
}
.calculator-container .calculate-button:active {
transform: translateY(0);
}
.calculator-container .result-area {
background-color: #e9f7ff;
border: 1px solid #cce5ff;
border-radius: 8px;
padding: 20px;
margin-top: 25px;
}
.calculator-container .result-area h3 {
color: #0056b3;
margin-top: 0;
margin-bottom: 15px;
font-size: 22px;
text-align: center;
}
.calculator-container .result-area p {
margin-bottom: 8px;
color: #333;
font-size: 16px;
}
.calculator-container .result-area p strong {
color: #007bff;
}
.calculator-container .result-area .error {
color: #dc3545;
font-weight: bold;
text-align: center;
}
Understanding the Quadratic Equation Solver
Welcome to our advanced calculator, specifically designed to solve quadratic equations. A quadratic equation is a fundamental concept in algebra, appearing in various fields from physics and engineering to economics and computer science. This tool helps you quickly find the roots (solutions) of any quadratic equation.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where:
x represents the unknown variable.
a, b, and c are coefficients, which are known numbers.
a cannot be equal to zero (if a=0, the equation becomes linear).
The "roots" of the equation are the values of x that satisfy the equation, i.e., make the equation true.
The Quadratic Formula
The roots of a quadratic equation can be found using the quadratic formula, a powerful tool derived by completing the square:
x = [-b ± √(b² - 4ac)] / 2a
This formula provides up to two solutions for x, depending on the value of the discriminant.
Understanding the Discriminant (b² – 4ac)
The term inside the square root, b² - 4ac, is called the discriminant (often denoted by Δ or D). Its value determines the nature of the roots:
If Discriminant > 0: The equation has two distinct real roots. This means there are two different numerical solutions for x that are real numbers.
If Discriminant = 0: The equation has exactly one real root (sometimes called a repeated or double root). Both solutions from the formula will be identical.
If Discriminant < 0: The equation has two complex conjugate roots. This means the solutions involve imaginary numbers (i, where i² = -1) and will appear in pairs like p + qi and p - qi.
How to Use the Calculator
Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember, if a term is missing, its coefficient is 0 (e.g., in x² - 4 = 0, b=0). If a term has no number in front, its coefficient is 1 (e.g., in x² + 2x + 1 = 0, a=1).
Enter Values: Input these numerical coefficients into the respective fields: 'Coefficient a', 'Coefficient b', and 'Coefficient c'.
Calculate: Click the "Calculate Roots" button.
View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex.
Examples of Quadratic Equations and Their Solutions
Let's look at some practical examples:
Example 1: Two Distinct Real Roots
Consider the equation: x² - 3x + 2 = 0
a = 1
b = -3
c = 2
Using the calculator with these values will yield:
Root 1 (x₁): 2.0000
Root 2 (x₂): 1.0000
This is because (x-1)(x-2) = x² – 3x + 2.
Example 2: One Real Root (Repeated)
Consider the equation: x² - 4x + 4 = 0
a = 1
b = -4
c = 4
The calculator will show:
Root (x): 2.0000
This is because (x-2)² = x² – 4x + 4.
Example 3: Two Complex Conjugate Roots
Consider the equation: x² + 2x + 5 = 0
a = 1
b = 2
c = 5
The calculator will output:
Root 1 (x₁): -1.0000 + 2.0000i
Root 2 (x₂): -1.0000 – 2.0000i
Here, the discriminant is negative (2² – 4*1*5 = 4 – 20 = -16), leading to complex solutions.
This advanced quadratic equation solver is a valuable tool for students, educators, and professionals who need to quickly and accurately find the roots of quadratic equations without manual calculation.