X-Intercept
The x-intercept is the point where a line or curve crosses the x-axis, found by setting y equal to zero in the equation.
Formula
\text{Set } y = 0, \text{ solve for } x
Definition
The x-intercept is the point where a line or curve crosses the x-axis, the value of $x$ when $y$ equals zero, found by substituting $y = 0$ into the equation and solving for $x$. A line may have zero, one, or (if it is the x-axis itself) infinitely many x-intercepts. More generally, the x-intercept is a zero of a function $f$, meaning $f(x) = 0$; for polynomials, the Fundamental Theorem of Algebra guarantees $n$ zeros (counting multiplicity) in the complex numbers for a degree-$n$ polynomial, though real x-intercepts correspond only to the real ones among them.
Example
In $y = 2x - 6$, setting $y = 0$ gives $2x = 6$, so $x = 3$ and the line crosses the x-axis at $(3, 0)$. For $3x + 6y = 12$, setting $y = 0$ gives $x = 4$. The cubic $f(x) = x^3 - 6x^2 + 11x - 6$ factors as $(x-1)(x-2)(x-3)$, giving x-intercepts at $x = 1, 2, 3$.
Key Insight
The x-intercept is where something "runs out" or "breaks even": on a graph of money over time, it is the day the balance hits zero. For quadratic and higher-degree functions, x-intercepts are the real roots or zeros, and locating them is one of the most important skills in algebra; numerical methods like Newton's method and bisection are dedicated entirely to finding them, a task that extends to locating equilibria in differential equations.