Quadratic equation solver — roots, vertex and discriminant
Solve ax² + bx + c = 0 for its roots — real or complex — with the discriminant, vertex, axis of symmetry and factored form shown.
Runs entirely in your browser — nothing is uploaded.
When the discriminant is negative the roots are complex conjugates — the parabola never crosses the x-axis.
How to use
- Enter a, b and c. The coefficients from ax² + bx + c = 0.
- Read the roots. Two real, one repeated, or a complex conjugate pair.
- Check the discriminant. Its sign tells you which case you are in before you look at the roots.
Questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is the discriminant, and it determines everything about the roots.
What does the discriminant tell me?
Positive means two distinct real roots and the parabola crosses the x-axis twice. Zero means one repeated root and it just touches. Negative means no real roots — the parabola misses the axis entirely and the solutions are complex.
What are complex roots?
When the discriminant is negative, the square root of a negative number gives an imaginary component. The roots come as a conjugate pair, a + bi and a − bi. They are perfectly valid solutions, just not points on the real graph.
What is the vertex?
The turning point, at x = −b ÷ 2a. It is the minimum if a is positive and the maximum if a is negative, and it sits on the axis of symmetry — the roots are always equidistant from it.