Enter the coefficients (a, b, c) for the quadratic equation ax² + bx + c = 0.
Results:
Understanding the Quadratic Formula and Its Graph
A quadratic equation is a polynomial equation of the second degree, 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 a must not be zero (a ≠ 0). If a = 0, the equation becomes linear, not quadratic.
The Quadratic Formula
The quadratic formula is a powerful tool used to find the roots (or solutions) of a quadratic equation. The roots are the values of x that make the equation true. The formula is derived from the process of completing the square on the general quadratic equation and is given by:
x = [-b ± √(b² - 4ac)] / 2a
The term inside the square root, b² - 4ac, is called the discriminant (often denoted by Δ or D). The discriminant tells us 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 (a repeated root).
If b² - 4ac < 0: There are two complex conjugate roots (no real roots).
Graphing Quadratic Equations
When you graph a quadratic equation of the form y = ax² + bx + c, you get a parabola. The parabola is a U-shaped curve. The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola (where the graph crosses the x-axis, i.e., where y = 0).
The Vertex of the Parabola
The vertex is the highest or lowest point on the parabola. It's a critical feature for understanding the shape and position of the graph. The coordinates of the vertex (h, k) can be found using the coefficients:
The x-coordinate of the vertex, h, is given by: h = -b / 2a
The y-coordinate of the vertex, k, is found by substituting this value of h back into the equation: k = a(h)² + b(h) + c
The vertex provides information about the axis of symmetry (the vertical line x = h) and the minimum or maximum value of the quadratic function.
Use Cases
Quadratic equations and their graphs are fundamental in many areas of science, engineering, and mathematics:
Physics: Describing projectile motion (the path of a ball thrown in the air is a parabola), calculating the trajectory of objects under gravity.
Engineering: Designing parabolic reflectors (like satellite dishes), optimizing shapes in architecture and structural design.
Economics: Modeling cost and revenue functions, finding points of maximum profit or minimum cost.
Optimization Problems: Finding maximum or minimum values in various scenarios.
This calculator helps you quickly find the roots and vertex of any quadratic equation, providing insights into its behavior and graphical representation.
function calculateQuadratic() {
var a = parseFloat(document.getElementById("coeffA").value);
var b = parseFloat(document.getElementById("coeffB").value);
var c = parseFloat(document.getElementById("coeffC").value);
var errorMessageDiv = document.getElementById("errorMessage");
errorMessageDiv.innerHTML = ""; // Clear previous errors
// Validate inputs
if (isNaN(a) || isNaN(b) || isNaN(c)) {
errorMessageDiv.innerHTML = "Please enter valid numbers for all coefficients.";
return;
}
if (a === 0) {
errorMessageDiv.innerHTML = "Coefficient 'a' cannot be zero for a quadratic equation.";
return;
}
// Calculate Discriminant
var discriminant = (b * b) – (4 * a * c);
var discriminantDiv = document.getElementById("discriminant");
discriminantDiv.innerHTML = "Discriminant (Δ): " + discriminant.toFixed(4);
var rootsDiv = document.getElementById("roots");
var vertexDiv = document.getElementById("vertex");
// Calculate Roots
var roots = [];
var rootsHTML = "Roots (x): ";
if (discriminant >= 0) {
var sqrtDiscriminant = Math.sqrt(discriminant);
var root1 = (-b + sqrtDiscriminant) / (2 * a);
var root2 = (-b – sqrtDiscriminant) / (2 * a);
if (root1 === root2) {
roots.push(root1);
rootsHTML += root1.toFixed(4) + " (repeated root)";
} else {
roots.push(root1);
roots.push(root2);
rootsHTML += root1.toFixed(4) + ", " + root2.toFixed(4);
}
} else {
// Complex roots
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(-discriminant) / (2 * a);
rootsHTML += realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i, ";
rootsHTML += realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i";
}
rootsDiv.innerHTML = rootsHTML;
// Calculate Vertex
var vertexX = -b / (2 * a);
var vertexY = a * Math.pow(vertexX, 2) + b * vertexX + c;
vertexDiv.innerHTML = "Vertex (h, k): (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")";
}
function resetCalculator() {
document.getElementById("coeffA").value = "";
document.getElementById("coeffB").value = "";
document.getElementById("coeffC").value = "";
document.getElementById("roots").innerHTML = "";
document.getElementById("vertex").innerHTML = "";
document.getElementById("discriminant").innerHTML = "";
document.getElementById("errorMessage").innerHTML = "";
}