A quadratic equation has the form ax² + bx + c = 0. Its graph is a parabola — a U-shaped curve that opens upward (when a > 0) or downward (when a < 0).

There are three main methods for solving quadratic equations: factoring, completing the square, and the quadratic formula. Factoring works when the expression can be written as a product of two binomials. For example, x² − 5x + 6 = (x − 2)(x − 3), giving solutions x = 2 and x = 3.

The quadratic formula x = (−b ± √(b² − 4ac)) / 2a works for any quadratic equation. The discriminant (b² − 4ac) tells you how many real solutions exist: positive means two solutions, zero means one, negative means no real solutions.

The vertex of the parabola is at x = −b/(2a). This gives the maximum or minimum value of the function.