Quadratic Formula Calculator
Solve any quadratic equation with real or complex roots, vertex and discriminant.
About Quadratic Formula Calculator
The quadratic formula solves ax² + bx + c = 0 for any real coefficients. The discriminant b² − 4ac determines the nature of the roots: positive gives two distinct real roots, zero gives one repeated root, negative gives two complex conjugate roots. The vertex is the highest or lowest point of the parabola y = ax² + bx + c.
Formulas
Quadratic formula
x = (−b ± √(b²−4ac)) / (2a)
- a
- Coefficient of x²
- b
- Coefficient of x
- c
- Constant term
- b²−4ac
- Discriminant D
Vertex
Vertex = (−b/(2a), −D/(4a))
Worked example
With the default values below, Quadratic Formula Calculator returns Root 1 (x₁) of 3. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (coefficient of x²)
- 1
- b (coefficient of x)
- -5
- c (constant term)
- 6
Results
- Root type
- two distinct real roots
- Root 1 (x₁)
- 3
- Root 2 (x₂)
- 2
- Discriminant (D = b²−4ac)
- 1.000000
- Axis of symmetry (x = −b/2a)
- 2.500000
- Vertex x-coordinate
- 2.500000
- Vertex y-coordinate
- -0.250000
Step by step
- 1Identify coefficients= a = 1, b = -5, c = 6
- 2Discriminant= D = 1
- 3√D= √1 = 1
- 4Root 1= x₁ = 3
- 5Root 2= x₂ = 2
- 6Vertex= (2.5, -0.25)
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What does the discriminant tell me?
The discriminant D = b² − 4ac reveals the nature of the roots without solving: D > 0 means two real roots; D = 0 means exactly one real root (the parabola touches the x-axis at its vertex); D < 0 means no real roots — the parabola doesn't cross the x-axis and the roots are complex.
What are complex roots?
When D < 0, the roots involve √(negative number) = imaginary numbers. They appear in conjugate pairs: a + bi and a − bi. They are mathematically valid but correspond to no real x-axis crossing.
How do I verify the roots?
Substitute each root back into ax² + bx + c. The result should be 0 (or very close to 0 due to floating-point precision). You can also verify by multiplying the factored form: a(x − x₁)(x − x₂) should expand to ax² + bx + c.