The quadratic equation is a fundamental concept in algebra. It is a second-degree polynomial equation, meaning it contains at least one term that is squared. The general form of a quadratic equation is:
ax² + bx + c = 0
where:
a, b, and c are coefficients, and they are real numbers.
a cannot be zero (if a were 0, the equation would become a linear equation: bx + c = 0).
x represents the unknown variable we are trying to solve for.
The Quadratic Formula
The solutions to a quadratic equation are called its roots. There can be zero, one, or two distinct real roots. The most common way to find these roots is by using the quadratic formula:
x = [-b ± √(b² – 4ac)] / 2a
The Discriminant
The term under the square root in the quadratic formula, b² - 4ac, is called the discriminant (often denoted by the Greek letter delta, Δ). The discriminant provides crucial information about the nature of the roots:
If b² - 4ac > 0: There are two distinct real roots.
If b² - 4ac = 0: There is exactly one real root (sometimes called a repeated or double root).
If b² - 4ac < 0: There are no real roots. Instead, there are two complex conjugate roots.
How the Calculator Works
This calculator takes your input for the coefficients a, b, and c. It then applies the quadratic formula:
It first calculates the discriminant (Δ = b² – 4ac).
Based on the discriminant's value, it determines whether there will be real roots, a single real root, or complex roots.
If real roots exist, it calculates them using the formula:
Root 1: (-b + √Δ) / 2a
Root 2: (-b – √Δ) / 2a
The results are then displayed clearly.
Use Cases
Quadratic equations and their solutions are not just abstract mathematical concepts. They appear in many real-world scenarios, including:
Physics: Calculating projectile motion (e.g., the trajectory of a ball), determining time to reach maximum height or return to the ground.
Economics: Modeling cost and revenue functions to find break-even points or profit maximization.
Geometry: Solving problems involving areas and shapes where dimensions are related quadratically.
Understanding how to solve quadratic equations is a foundational skill in mathematics and science, enabling us to model and solve a wide range of problems.
function calculateRoots() {
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("result");
// Clear previous results and error messages
resultDiv.innerHTML = ";
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = 'Error: Please enter valid numbers for all coefficients.';
return;
}
if (a === 0) {
resultDiv.innerHTML = 'Error: Coefficient "a" cannot be zero for a quadratic equation. This is a linear equation.';
return;
}
var discriminant = b * b – 4 * a * c;
var roots = [];
if (discriminant > 0) {
var x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
var x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
roots.push(x1.toFixed(4)); // Format to 4 decimal places
roots.push(x2.toFixed(4));
resultDiv.innerHTML = 'There are two distinct real roots:x₁ = ' + roots[0] + 'x₂ = ' + roots[1] + ";
} else if (discriminant === 0) {
var x = -b / (2 * a);
roots.push(x.toFixed(4));
resultDiv.innerHTML = 'There is exactly one real root (a repeated root):x = ' + roots[0] + ";
} else {
// Complex roots case
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(-discriminant) / (2 * a);
roots.push(realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i");
roots.push(realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i");
resultDiv.innerHTML = 'There are two complex conjugate roots:x₁ = ' + roots[0] + 'x₂ = ' + roots[1] + ";
}
}
function resetForm() {
document.getElementById("coefficientA").value = "";
document.getElementById("coefficientB").value = "";
document.getElementById("coefficientC").value = "";
document.getElementById("result").innerHTML = 'Enter coefficients a, b, and c to find the roots.';
}