All calculatorsQuadratic Equation Calculator
Solve any quadratic equation ax² + bx + c = 0 with exact answers (fractions / radicals), decimal approximations, and a clean mini graph + key features (vertex, axis of symmetry, discriminant). Includes optional step-by-step.
Background
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. The solutions come from the quadratic formula. The discriminant D = b² − 4ac tells you whether the roots are real, repeated, or complex.
How to use this calculator
- Enter your coefficients a, b, c from ax² + bx + c = 0.
- Click Calculate to get exact roots, decimals, and key features (vertex, axis, discriminant).
- Use Quick picks to see common cases (two real roots, repeated root, complex roots).
How this calculator works
- Computes the discriminant: D = b² − 4ac.
- Uses the quadratic formula: x = (−b ± √D) / (2a).
- Finds vertex: xᵥ = −b/(2a), yᵥ = f(xᵥ), and the axis of symmetry x = xᵥ.
- If factoring is “nice,” it also shows a factor form like (x − r₁)(x − r₂) (or (x − r)²).
Formula & Equation Used
Discriminant: D = b² − 4ac
Quadratic formula: x = (−b ± √D) / (2a)
Vertex: xᵥ = −b/(2a), yᵥ = a xᵥ² + b xᵥ + c
Axis of symmetry: x = −b/(2a)
Example Problems & Step-by-Step Solutions
Example 1 — Two real roots
Solve x² − 6x + 8 = 0.
- Compute D: D = (−6)² − 4(1)(8) = 36 − 32 = 4.
- Apply formula: x = (6 ± √4)/2 = (6 ± 2)/2.
- Roots: x = 2 and x = 4.
Example 2 — One repeated root
Solve x² − 4x + 4 = 0.
- D: D = (−4)² − 4(1)(4) = 16 − 16 = 0 → one repeated root.
- Root: x = −b/(2a) = 4/2 = 2.
- Factor form: (x − 2)².
Example 3 — Complex roots
Solve x² + 2x + 5 = 0.
- D: D = 2² − 4(1)(5) = 4 − 20 = −16.
- √D = √(−16) = 4i.
- Roots: x = (−2 ± 4i)/2 = −1 ± 2i.
Frequently Asked Questions
Q: What does the discriminant tell me?
If D > 0, you get two real roots. If D = 0, one repeated real root. If D < 0, two complex roots.
Q: What if a = 0?
Then it’s not a quadratic equation anymore — it becomes linear: bx + c = 0. This calculator will flag that.
Q: Will you always show factoring?
Only when it’s clean (like integer or simple rational roots). Otherwise, the quadratic formula is the reliable method.
Q: Why show both exact and decimal answers?
Exact answers are best for algebra and proofs. Decimals are useful for graphing and quick checking.
Q: What is the vertex?
It’s the turning point of the parabola. The x-coordinate is −b/(2a), and y is f(x) at that x.
