A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The solution is:
x = (-b ± √(b² - 4ac)) / 2a
The expression under the square root, D = b² - 4ac, is called the discriminant.
D > 0: Two distinct real roots. The parabola crosses the x-axis at two points.
D = 0: One real root (repeated). The parabola touches the x-axis at exactly one point — the vertex.
D < 0: Two complex conjugate roots. The parabola does not cross the x-axis. Roots are expressed as a ± bi where i = √(-1).
Besides the quadratic formula, equations can be solved by factoring (when the trinomial factors neatly), completing the square, or graphing. The quadratic formula is the universal method — it works for every quadratic equation regardless of whether it factors cleanly.