The sign of a function and quadratic inequalities

The sign of f(x) = ax² + bx + c follows the parabola: above the x-axis f is positive, below it f is negative, on it f is zero. The discriminant tells how the parabola meets the axis. If b² − 4ac > 0 there are two roots L < M. The parabola opens up when a > 0, so it is above the axis outside the roots and below it between them: f has the sign of a outside [L, M] and the opposite sign between L and M. If b² − 4ac = 0 the parabola only touches the axis at one point, so f has the sign of a everywhere except at that root, where it is zero. If b² − 4ac < 0 the parabola never meets the axis, so f has the sign of a for every real x. Example: f(x) = x² − 2x − 8 = (x + 2)(x − 4), roots −2 and 4, a = 1 > 0: positive on ]−∞, −2[ and on ]4, ∞[, negative on ]−2, 4[.