Enter the slope (m) and the y-intercept (b) to find where the line crosses the x-axis.
Quadratic Equation (ax² + bx + c = 0)
Enter the coefficients for the quadratic equation to find the x-intercepts (roots).
function calculateLinearX() {
var m = parseFloat(document.getElementById('linear_m').value);
var b = parseFloat(document.getElementById('linear_b').value);
var resDiv = document.getElementById('linear_result');
if (isNaN(m) || isNaN(b)) {
resDiv.innerHTML = "Please enter valid numbers for both m and b.";
resDiv.style.backgroundColor = "#fee2e2";
resDiv.style.color = "#b91c1c";
resDiv.style.display = "block";
return;
}
if (m === 0) {
if (b === 0) {
resDiv.innerHTML = "Result: Infinite x-intercepts (The line is the x-axis).";
} else {
resDiv.innerHTML = "Result: No x-intercept (The line is parallel to the x-axis).";
}
} else {
var x = -b / m;
resDiv.innerHTML = "X-Intercept: (" + x.toFixed(4) + ", 0)Calculation: 0 = " + m + "x + " + b + " → x = -" + b + "/" + m + "";
}
resDiv.style.backgroundColor = "#dcfce7";
resDiv.style.color = "#15803d";
resDiv.style.display = "block";
}
function calculateQuadraticX() {
var a = parseFloat(document.getElementById('quad_a').value);
var b = parseFloat(document.getElementById('quad_b').value);
var c = parseFloat(document.getElementById('quad_c').value);
var resDiv = document.getElementById('quad_result');
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resDiv.innerHTML = "Please enter valid numbers for a, b, and c.";
resDiv.style.backgroundColor = "#fee2e2";
resDiv.style.color = "#b91c1c";
resDiv.style.display = "block";
return;
}
if (a === 0) {
// It's actually a linear equation bx + c = 0
if (b === 0) {
resDiv.innerHTML = (c === 0) ? "Infinite intercepts." : "No intercepts.";
} else {
var x = -c / b;
resDiv.innerHTML = "Linear result: 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);
resDiv.innerHTML = "Two Real Intercepts:x₁ = " + x1.toFixed(4) + "x₂ = " + x2.toFixed(4);
} else if (discriminant === 0) {
var x = -b / (2 * a);
resDiv.innerHTML = "One Real Intercept (Vertex):x = " + x.toFixed(4);
} else {
resDiv.innerHTML = "No Real Intercepts: The discriminant is negative (" + discriminant.toFixed(2) + "), meaning the parabola does not cross the x-axis.";
}
}
resDiv.style.backgroundColor = "#dcfce7";
resDiv.style.color = "#15803d";
resDiv.style.display = "block";
}
Understanding the X-Intercept
In algebra and coordinate geometry, the x-intercept is the point where a graph crosses or touches the x-axis. At this specific point, the value of the y-coordinate is always zero (y = 0).
How to Find the X-Intercept Algebraically
Whether you are dealing with a simple straight line or a complex curve, the fundamental rule for finding the x-intercept remains the same:
Set the value of y to 0 in your equation.
Solve the resulting equation for x.
1. Linear Equations (Slope-Intercept Form)
For a line represented by y = mx + b:
Formula: x = -b / m
Example: Given y = 2x + 4. Set y to 0:
0 = 2x + 4
-4 = 2x
x = -2. The x-intercept is (-2, 0).
2. Quadratic Equations
For a parabola represented by ax² + bx + c = 0, you find the x-intercepts using the Quadratic Formula:
Formula: x = [-b ± sqrt(b² – 4ac)] / 2a
The number of x-intercepts depends on the discriminant (b² – 4ac):
Positive: Two distinct x-intercepts.
Zero: One x-intercept (the vertex touches the axis).
Negative: No real x-intercepts (the graph is entirely above or below the axis).
Real-World Example
Imagine a business model where profit (y) is calculated based on the number of units sold (x). The x-intercept in this context represents the break-even point—the exact number of units you need to sell to make exactly zero profit before you start earning a gain.
Common FAQ
Can a function have no x-intercept?
Yes. For example, the horizontal line y = 5 will never cross the x-axis because y is always 5 and never 0.
What is the difference between an x-intercept and a y-intercept?
The x-intercept is where y = 0 (crosses the horizontal axis), while the y-intercept is where x = 0 (crosses the vertical axis).