The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
This formula works for every quadratic equation. The ± sign means you get two solutions: one using + and one using −. When the discriminant (b² − 4ac) equals zero, both solutions are the same.
Understanding the Discriminant
| Discriminant (Δ = b²−4ac) | Root Type | Graph Behavior |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One repeated real root | Parabola touches x-axis at vertex |
| Δ < 0 | Two complex conjugate roots | Parabola doesn't cross x-axis |
Alternative Methods
Factoring: Works when the roots are "nice" integers or fractions. x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or x = 3. Fast when it works, but many quadratics don't factor neatly.
Completing the square: Rewrite ax² + bx + c as a(x − h)² + k. This gives the vertex form directly. The quadratic formula is actually derived from completing the square on the general form.
Vertex and Axis of Symmetry
Every parabola y = ax² + bx + c has a vertex at x = −b/(2a), y = c − b²/(4a). The axis of symmetry is the vertical line x = −b/(2a). If a > 0, the parabola opens upward (vertex is a minimum). If a < 0, it opens downward (vertex is a maximum).