Quadratic Formula Calculator
Solve any quadratic equation of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to get the solutions (real or complex), the discriminant, and the vertex of the parabola.
Solutions (x)
2 and 3
Check the highlighted fields to see your result.
- Equation
- x² − 5x + 6 = 0
- Type of roots
- Two real roots
- Discriminant (Δ)
- 1
- Vertex
- (2.5, −0.25)
Results are estimates for information only and do not constitute financial, medical, or legal advice.
How the quadratic formula works
The quadratic formula gives the values of x that make ax² + bx + c equal 0. The part under the square root, the discriminant, decides how many real solutions there are. The vertex is the highest or lowest point of the parabola y = ax² + bx + c.
The formula
x = (−b ± √(b² − 4ac)) ÷ 2a
Worked example
Solve x² − 5x + 6 = 0, so a = 1, b = −5, and c = 6.
- Discriminant: (−5)² − 4 × 1 × 6 = 25 − 24 = 1.
- x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2.
- Solutions: x = 2 and x = 3.
Frequently asked questions
What does the discriminant tell me?
The discriminant, b² − 4ac, tells you what kind of solutions there are before you solve. If it is positive, there are two real solutions; if it is 0, there is one (a double root); if it is negative, there are no real solutions, only two complex ones.
What are complex roots?
When the discriminant is negative, the square root of a negative number appears, which uses i (the square root of −1). The parabola never crosses the x-axis, so there is no real x that makes the equation true.
What if my equation is not in the form ax² + bx + c = 0?
Move every term to one side so the other side is 0, then read off a, b, and c. For example, 2x² = 3x − 1 becomes 2x² − 3x + 1 = 0, so a = 2, b = −3, and c = 1. If a term is missing, its coefficient is 0.