function calculateQuadraticRoots() {
var a = parseFloat(document.getElementById("coeffA").value);
var b = parseFloat(document.getElementById("coeffB").value);
var c = parseFloat(document.getElementById("coeffC").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients.";
return;
}
if (a === 0) {
resultDiv.innerHTML = "Coefficient 'a' cannot be zero for a quadratic equation.";
// If a=0, it's a linear equation: bx + c = 0 => x = -c/b
if (b !== 0) {
var linearRoot = -c / b;
resultDiv.innerHTML += "Since 'a' is 0, this is a linear equation. The root is x = " + linearRoot.toFixed(4) + "";
} else if (c !== 0) {
resultDiv.innerHTML += "This equation has no solution (e.g., 0 = 5).";
} else {
resultDiv.innerHTML += "This equation has infinite solutions (e.g., 0 = 0).";
}
return;
}
var discriminant = b * b – 4 * a * c;
if (discriminant > 0) {
var x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
var x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
resultDiv.innerHTML = "Two distinct real roots:x1 = " + x1.toFixed(4) + "x2 = " + x2.toFixed(4) + "";
} else if (discriminant === 0) {
var x = -b / (2 * a);
resultDiv.innerHTML = "One real root (repeated):x1 = x2 = " + x.toFixed(4) + "";
} else { // discriminant < 0 (complex roots)
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
resultDiv.innerHTML = "Two distinct complex roots:x1 = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix2 = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i";
}
}
Understanding Quadratic Equations and the Quadratic Formula
Quadratic equations are fundamental in pre-calculus and appear frequently in various fields, from physics to engineering and economics. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. Its standard form is:
ax² + bx + c = 0
where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. The solutions to this equation are called its roots, and they represent the x-intercepts of the parabola that the equation describes when graphed.
The Quadratic Formula
While some quadratic equations can be solved by factoring or completing the square, the quadratic formula provides a universal method to find the roots for any quadratic equation. The formula is:
x = [-b ± √(b² - 4ac)] / 2a
The term inside the square root, (b² - 4ac), is called the **discriminant**, often denoted by Δ (Delta). The discriminant tells us about the nature of the roots:
- If
Δ > 0: There are two distinct real roots. The parabola intersects the x-axis at two different points.
- If
Δ = 0: There is exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point (its vertex).
- If
Δ < 0: There are two distinct complex (non-real) roots. The parabola does not intersect the x-axis.
How to Use This Calculator
This calculator simplifies the process of finding the roots of a quadratic equation. Simply input the numerical values for the coefficients 'a', 'b', and 'c' from your equation ax² + bx + c = 0 into the respective fields. For example, if your equation is 2x² + 5x - 3 = 0, you would enter 2 for 'a', 5 for 'b', and -3 for 'c'. Click "Calculate Roots" to instantly see the solutions.
Examples:
Example 1: Two Distinct Real Roots
Equation: x² - 3x + 2 = 0
Discriminant = (-3)² - 4(1)(2) = 9 - 8 = 1 (positive)
Roots: x = [3 ± √1] / 2(1)
x1 = (3 + 1) / 2 = 2
x2 = (3 - 1) / 2 = 1
Calculator Input: a=1, b=-3, c=2
Calculator Output: x1 = 2.0000, x2 = 1.0000
Example 2: One Real Root (Repeated)
Equation: x² - 4x + 4 = 0
Discriminant = (-4)² - 4(1)(4) = 16 - 16 = 0
Roots: x = [4 ± √0] / 2(1)
x1 = x2 = 4 / 2 = 2
Calculator Input: a=1, b=-4, c=4
Calculator Output: x1 = x2 = 2.0000
Example 3: Two Complex Roots
Equation: x² + 2x + 5 = 0
Discriminant = (2)² - 4(1)(5) = 4 - 20 = -16 (negative)
Roots: x = [-2 ± √-16] / 2(1)
x = [-2 ± 4i] / 2
x1 = -1 + 2i
x2 = -1 - 2i
Calculator Input: a=1, b=2, c=5
Calculator Output: x1 = -1.0000 + 2.0000i, x2 = -1.0000 – 2.0000i
/* Basic styling for the calculator */
.calculator-container {
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
font-family: Arial, sans-serif;
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.calc-input-group {
margin-bottom: 15px;
}
.calc-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #555;
}
.calc-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
box-sizing: border-box;
}
.calculator-container button {
background-color: #007bff;
color: white;
padding: 12px 20px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 16px;
width: 100%;
margin-top: 10px;
}
.calculator-container button:hover {
background-color: #0056b3;
}
.calc-result {
margin-top: 20px;
padding: 15px;
border: 1px solid #e0e0e0;
border-radius: 4px;
background-color: #e9ecef;
min-height: 50px;
color: #333;
font-size: 1.1em;
font-weight: bold;
}
.calc-result p {
margin: 5px 0;
}
.calculator-article {
max-width: 600px;
margin: 40px auto;
font-family: Arial, sans-serif;
line-height: 1.6;
color: #333;
}
.calculator-article h3, .calculator-article h4 {
color: #007bff;
margin-top: 25px;
margin-bottom: 15px;
}
.calculator-article p {
margin-bottom: 10px;
}
.calculator-article ul {
list-style-type: disc;
margin-left: 20px;
margin-bottom: 10px;
}
.calculator-article code {
background-color: #eee;
padding: 2px 4px;
border-radius: 3px;
font-family: 'Courier New', Courier, monospace;
}