Use this calculator to find the roots (solutions) of a quadratic equation in the standard form: ax² + bx + c = 0.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of the second degree. The general form is ax² + bx + c = 0, where 'x' represents an unknown, and 'a', 'b', and 'c' are coefficients, with 'a' not equal to zero for it to be a true quadratic equation. The solutions to a quadratic equation are called its roots.
The Quadratic Formula
The roots of a quadratic equation can be found using the quadratic formula:
x = [-b ± √(b² - 4ac)] / 2a
The term (b² - 4ac) is known as the discriminant (often denoted by Δ or D). The value of the discriminant determines the nature of the roots:
If Δ > 0: There are two distinct real roots.
If Δ = 0: There is exactly one real root (a repeated root).
If Δ < 0: There are two complex conjugate roots.
Examples of Quadratic Equations
Let's look at some examples:
Equation: x² - 3x + 2 = 0
Here, a = 1, b = -3, c = 2.
Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1.
Since Δ > 0, there are two distinct real roots.
x = [3 ± √1] / 2(1)
x1 = (3 + 1) / 2 = 2
x2 = (3 – 1) / 2 = 1
Equation: x² + 4x + 4 = 0
Here, a = 1, b = 4, c = 4.
Discriminant = (4)² – 4(1)(4) = 16 – 16 = 0.
Since Δ = 0, there is one real root.
x = [-4 ± √0] / 2(1)
x = -4 / 2 = -2
Equation: x² + x + 1 = 0
Here, a = 1, b = 1, c = 1.
Discriminant = (1)² – 4(1)(1) = 1 – 4 = -3.
Since Δ < 0, there are two complex conjugate roots.
x = [-1 ± √-3] / 2(1)
x1 = (-1 + i√3) / 2
x2 = (-1 – i√3) / 2
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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("quadraticResult");
resultDiv.className = "calculator-result"; // Reset class for potential error states
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Please enter valid numbers for all coefficients.";
resultDiv.classList.add("error");
return;
}
var output = "";
if (a === 0) {
// This is a linear equation: bx + c = 0
if (b === 0) {
if (c === 0) {
output = "Infinite solutions (0 = 0).";
} else {
output = "No solution (e.g., " + c + " = 0, which is false).";
resultDiv.classList.add("error");
}
} else {
var x = -c / b;
output = "This is a linear equation (a=0). Solution: x = " + x.toFixed(4);
}
} else {
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);
output = "Two distinct real roots:x1 = " + x1.toFixed(4) + "x2 = " + x2.toFixed(4);
} else if (discriminant === 0) {
var x = -b / (2 * a);
output = "One real root (repeated root):x = " + x.toFixed(4);
} else {
// discriminant < 0, complex roots
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a);
output = "Two complex conjugate roots:x1 = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix2 = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i";
}
}
resultDiv.innerHTML = output;
}