Quadratic Equation Calculator
Enter a, b and c. Get the roots, vertex, factors, steps and a graph.
Calculate
How to use this calculator
Enter the numbers a, b and c from ax² + bx + c = 0. a cannot be 0. The roots, the working, the vertex and the graph update instantly.
How it's calculated
The discriminant is D = b² − 4ac. The roots are x = (−b ± √D) / 2a. If D is positive there are two real roots, if D = 0 there is one repeated root, and if D is negative there are two complex roots. The vertex is at x = −b / 2a.
Worked example
x² − 5x + 6 = 0 has a = 1, b = −5, c = 6. D = 25 − 24 = 1, so x = (5 ± 1) / 2, giving x = 2 or x = 3.
Frequently asked questions
What does the discriminant tell me?
It tells you how many real roots the equation has: positive means two, zero means one, negative means none (the parabola never touches the x-axis).
Why is my answer left with a square root?
When the root is not a whole number or fraction, the exact value is kept, like 1 ± √5, and a decimal is shown too.
How do I solve by factorising?
If the roots are rational the Factorised line shows the factors, for example (x − 2)(x − 3) = 0.
Related tools
Last updated October 2026. Results are for learning and guidance; always check important work. Calculations run in your browser and are not stored.