Quadratic Formula Calculator
Enter the coefficients a, b and c of a quadratic equation. The calculator finds both solutions, including complex ones, and describes the parabola.
Result
x = (−b ± √(b² − 4ac)) ÷ (2a) b² − 4ac = (-5)² − 4 × 1 × 6 = 1 x = (5 ± 1) ÷ 2
- Nature of the roots
- Two distinct real roots
- Discriminant (b² − 4ac)
- 1
- Vertex
- (2.5, -0.25)
- Axis of symmetry
- x = 2.5
- Opens
- Upward, so the vertex is a minimum
- y-intercept
- 6
Formula
- x = (−b ± √(b² − 4ac)) ÷ (2a)
- Discriminant Δ = b² − 4ac
- Vertex = (−b ÷ 2a, c − b² ÷ 4a)
How the Quadratic Formula Calculator works
The quadratic formula solves any equation of the form ax² + bx + c = 0. It comes from completing the square and always works, whether or not the expression factors neatly.
The discriminant tells you what kind of answers to expect. Positive means two real solutions, zero means one repeated solution, and negative means two complex solutions.
Worked example
For x² − 5x + 6 = 0: the discriminant is (−5)² − 4 × 1 × 6 = 1. Then x = (5 ± 1) ÷ 2, so x = 3 or x = 2. The vertex is at (2.5, −0.25).
Frequently asked questions
What does a negative discriminant mean?
The parabola does not cross the x-axis, so there are no real solutions. The two solutions are complex numbers that are conjugates of each other.
How do I find the vertex of a parabola?
The x-coordinate is −b ÷ 2a. Substitute it into the equation to get the y-coordinate.
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