Zeros (Roots) of a Quadratic Polynomial

A root (also called a zero) of the quadratic equation $p(x) = ax^2 + bx + c = 0$ is a real number $\alpha$ such that substituting $x = \alpha$ into $p(x)$ makes the expression equal to zero, i.e. $p(\alpha) = a\alpha^2 + b\alpha + c = 0$.

Geometrically (though this syllabus does not require the graph), a root is a value of $x$ at which the polynomial's value crosses zero. Algebraically, "finding the roots" simply means finding every real value of $x$ that satisfies the equation.

A quadratic equation, because it has degree 2, has at most two roots (it may have exactly two distinct real roots, exactly one repeated real root, or no real root at all — see §5 on the discriminant). This is different from a linear equation, which always has exactly one root.