Quadratic Solver
Roots, discriminant and vertex for axยฒ + bx + c โ complex roots included.
About this solver
Enter a, b and c and read everything at once: the equation written out, both roots (real or complex), the discriminant with what it means, the quadratic formula with your numbers substituted, and the vertex. Set a to zero and it politely solves the linear equation instead.
The discriminant decides everything
Before the formula finishes, bยฒ โ 4ac has already told the story. Positive: the parabola crosses the x-axis twice โ two real roots. Zero: it just kisses the axis โ one repeated root. Negative: it never arrives, and the roots become a complex-conjugate pair (x = p ยฑ qi) โ which isn't failure, just the algebra reporting that no real number squares to a negative. Geometry ties it together: the vertex sits at x = โb/2a, exactly midway between the roots when they exist, and a's sign says whether the parabola opens up (vertex is the minimum) or down (maximum). Quadratics earn their place in the curriculum by being everywhere: thrown-ball trajectories, profit curves, area problems where length ร width meets a fixed perimeter, and the compound growth of two periods.
Frequently asked questions
What is the quadratic formula?
x = (โb ยฑ โ(bยฒ โ 4ac)) / 2a โ it solves any equation of the form axยฒ + bx + c = 0, and the ยฑ is why there are generally two roots.
What does the discriminant tell you?
bยฒ โ 4ac decides the root count before you finish the formula: positive means two real roots, zero one repeated root, negative a complex pair โ the parabola never touches the x-axis.
What are complex roots?
When bยฒ โ 4ac is negative, its square root involves i (โโ1), giving roots like 2 ยฑ 3i. They're the honest algebraic answer to a parabola with no real crossings, and they always come as a mirrored pair.
How do I find the vertex of a parabola?
x = โb / 2a, then substitute back for y. It's the parabola's minimum when a is positive (opens up) and maximum when a is negative โ and it sits exactly midway between the roots when they exist.