Quadratic Equation Solver
Enter the coefficients a, b and c to find the roots of a quadratic equation, the discriminant, and the vertex of the parabola.
Result
Enter your values and select Calculate to see the result.
How to use this calculator
- Enter a, b and c from your equation in the form ax² + bx + c = 0. Move every term to one side first.
- Coefficients can be negative or decimal. a cannot be 0.
- The result shows the roots and the shape of the parabola.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant D = b² − 4ac decides the type of roots:
- D > 0: two different real roots
- D = 0: one repeated real root
- D < 0: two complex roots −b/2a ± (√−D / 2a)i
The vertex is at x = −b ÷ 2a. The calculator uses a numerically stable version of the formula to avoid rounding errors when b² is much larger than 4ac.
Worked example
x² − 5x + 6 = 0
D = 25 − 24 = 1, so there are two real roots: x = (5 ± 1) ÷ 2, giving x = 2 and x = 3.
2x² + 4x + 5 = 0
D = 16 − 40 = −24, so the roots are complex: x = −1 ± 1.2247i. The vertex is (−1, 3), above the x-axis.
Assumptions and limitations
- Results are shown to up to 8 decimal places.
- If a = 0 the equation is linear (bx + c = 0), and the calculator asks you to check it.
Frequently asked questions
How do I get my equation into ax² + bx + c form?
Expand any brackets and move every term to one side so the other side is 0. For example 2x² = 3x + 5 becomes 2x² − 3x − 5 = 0, so a = 2, b = −3, c = −5.
What do complex roots mean graphically?
The parabola never crosses the x-axis, so there are no real solutions.