Quadratic Formula — ax² + bx + c = 0
x = (−b ± √(b²−4ac)) / 2a
Variable Key
Quadratic formula
Finds both roots simultaneously.
Discriminant analysis
Determines the nature of roots before solving.
Vieta's formulas
Relationships between roots and coefficients.
Vertex form
Rewrite in vertex form to find the parabola's turning point.
The quadratic formula solves any equation of the form ax² + bx + c = 0 for x. It works for all quadratics — even ones that cannot be factored — making it the most universal solving method. The formula was known to Babylonian mathematicians as early as 2000 BC.
- 1Arrange the equation in standard form: ax² + bx + c = 0
- 2Identify a (coefficient of x²), b (coefficient of x), c (constant)
- 3Calculate the discriminant: Δ = b² − 4ac
- 4If Δ ≥ 0: substitute into x = (−b ± √Δ) / 2a for two real roots
- 5If Δ < 0: the equation has two complex (non-real) roots
When a = 0
The equation becomes linear (bx + c = 0), not quadratic. Solve as x = −c/b.
Vertex of the parabola
The x-coordinate of the vertex is x = −b/(2a), the midpoint of the two roots.
| Δ value | Number of real roots | Graph crosses x-axis |
|---|---|---|
| Δ > 0 | Two distinct real roots | At two points |
| Δ = 0 | One repeated real root | Touches at one point |
| Δ < 0 | No real roots (complex) | Does not cross |
Fun Fact
The quadratic formula was first written in modern algebraic notation by René Descartes in 1637. Before that, mathematicians described the same method in words and geometric diagrams.