The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Example: Solve 2x² − 4x − 6 = 0. Here a = 2, b = −4, c = −6. The discriminant is (−4)² − 4 × 2 × (−6) = 16 + 48 = 64, and √64 = 8. So x = (4 ± 8) ÷ 4, giving x = 3 and x = −1. The vertex is at x = 4 ÷ 4 = 1, y = 2 − 4 − 6 = −8, so the point (1, −8) is the minimum.
Types of solutions
| Discriminant | Roots | Graph |
|---|---|---|
| D > 0 | Two distinct real roots | Crosses the x-axis twice |
| D = 0 | One repeated real root | Touches the x-axis at the vertex |
| D < 0 | Two complex roots p ± qi | Does not reach the x-axis |
Sum and product of roots (Vieta’s formulas): x₁ + x₂ = −b/a and x₁ × x₂ = c/a — a quick way to check an answer.
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a.
What does the discriminant tell me?
The discriminant D = b² − 4ac determines the type of roots. If D > 0 there are two different real roots; if D = 0 there is one repeated real root; if D < 0 there are two complex conjugate roots and the parabola does not cross the x-axis.
What is the vertex of a parabola?
The vertex is the turning point of y = ax² + bx + c, at x = −b ÷ 2a. It is the minimum when a > 0 and the maximum when a < 0.
Can I solve by factoring instead?
Yes, when the equation factors nicely. x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or 3. The quadratic formula works for every quadratic, including those that do not factor over the integers.