How it works
Enter the coefficients of ax² + bx + c = 0. The quadratic formula gives the roots: the x-values where the parabola crosses the x-axis.
The discriminant b² − 4ac tells you what kind of roots to expect: positive means two real roots, zero means one repeated root, and negative means two complex roots of the form p ± qi.
The vertex is the turning point of the parabola, at x = −b/2a. The calculator uses a numerically stable form of the formula so small roots stay accurate.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a; vertex h = −b/2a, k = c − b²/4a
Questions people ask
What does a negative discriminant mean?
The parabola never touches the x-axis, so there are no real solutions. The roots are complex: for x² + 2x + 5 = 0 they are −1 ± 2i.
How do I find the vertex of a parabola?
h = −b ÷ 2a, then substitute h back into the equation to get k. For x² − 5x + 6 the vertex is (2.5, −0.25).
Can I solve a quadratic by factoring instead?
Yes, when it factors nicely: x² − 5x + 6 = (x − 2)(x − 3), so x = 2 or 3. The formula works for every quadratic, including ones that don't factor.