Solution state
The equation has two distinct real roots.
Mathematics · Algebra
Quadratic equation solver
Find real or complex roots of ax² + bx + c = 0, with discriminant, vertex, symmetry axis and a live parabola.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Root x₁
First root, or the unique double root.
Root x₂
Distinct real root or complex conjugate.
Discriminant
Determines the real or complex root case.
Vertex
Turning point of the parabola.
Factorized form
Factorization over the real numbers.
Symmetry axis x
Vertical line through the vertex.
Opening direction
Determined by the sign of a.
Results show up to four significant digits for readability; internal calculations retain full precision.
See calculation detailsExpansion, discriminant and step-by-step solution…
01Formulas and symbols
Formulas used
Δ
Δ = b² − 4ac; x = (−b ± √Δ)/(2a)Δ: discriminant; x₁ and x₂: equation roots.S
S = (−b/(2a), c − b²/(4a))S: vertex; xₛ: vertex x-coordinate.| Symbol | Meaning |
|---|---|
| a | a — coefficient of x² |
| b | b — coefficient of x |
| c | c — constant term |
| Δ | Discriminant |
| x₁ | Root x₁ |
| x₂ | Root x₂ |
| S | Vertex |
| xₛ | Symmetry axis x |
| f(x) | Factorized form |
02Validation example
Reference numerical case
- a = 1, b = −3, c = 2: Δ = 1, roots 1 and 2, vertex (1.5, −0.25). For (1, 2, 1) the double root is −1; for (1, 0, 1) the roots are ±i.
03How it works
For Δ > 0 there are two real roots; Δ = 0 gives one double root. When Δ < 0 the conjugate roots have imaginary parts ±√(−Δ)/(2|a|). The graph then has no real-axis intersection.
S is the vertex: its x coordinate defines the symmetry axis, and its y coordinate is the minimum for a > 0 or maximum for a < 0.
04Limits and conventions
Real coefficients and a ≠ 0. This solves a polynomial of degree two, not a general nonlinear equation. Floating-point arithmetic cannot reliably distinguish a repeated root from extremely close roots beyond machine precision.
05Common mistakes
A negative discriminant is not an invalid equation: it means complex roots. Changing the sign of a changes the opening direction.
06Practical questions
What does a negative discriminant mean?
A negative discriminant means the quadratic has no real roots. It has two complex-conjugate roots whose imaginary parts have opposite signs.
When does a quadratic equation have one repeated root?
A quadratic has one repeated real root when the discriminant Δ = b² − 4ac equals zero. The repeated root is x = −b/(2a), which is also the x-coordinate of the parabola’s vertex.
07Technical references
- Precalculus 2e — quadratic functions — OpenStax