Solve quadratic equations — roots, discriminant, vertex and step-by-step working.
Solve ax² + bx + c = 0. Enter a, b and c — a must not be zero.
Roots (solutions for x)
How to use this tool
Write your equation in the form ax² + bx + c = 0 and read off a, b and c.
Type a, b and c into the calculator — a must not be zero.
Read the two roots, the discriminant and the vertex, with the working shown.
Use the vertex form for the turning point and the factored form to check the roots.
Formula & worked example
x² − 3x + 2 = 0 has discriminant Δ = 9 − 8 = 1, so two real roots: x = 2 and x = 1. Its vertex is (1.5, −0.25), its vertex form is y = (x − 1.5)² − 0.25, and it factors as (x − 2)(x − 1) = 0.
Limitations — read this before you rely on it
This solves single-variable quadratic equations with real coefficients only. It does not solve cubic or higher-degree equations, simultaneous equations, or inequalities, and complex roots are shown in a + bi form without a graph. Roots are rounded to 4 decimal places, so for exact surd answers (like √2) the displayed value is an approximation.
Frequently asked questions
Enter a, b and c for ax² + bx + c = 0. The calculator gives both roots, the discriminant, the vertex, the axis of symmetry, the vertex form and the factored form where the roots are real.
x = (−b ± √(b² − 4ac)) ÷ 2a. It comes from completing the square and works for every quadratic equation, including ones that do not factor neatly.
The discriminant is Δ = b² − 4ac. If Δ > 0 there are two distinct real roots, if Δ = 0 there is one repeated root, and if Δ < 0 there are no real roots — the two roots are complex.
Yes. For example, x² + 1 = 0 has roots i and −i, where i² = −1. The calculator shows complex roots in a + bi form.
The vertex form is y = a(x − h)² + k, where (h, k) is the vertex. It makes the turning point and the axis of symmetry (x = h) easy to read, and shows whether the parabola opens up (a > 0, a minimum) or down (a < 0, a maximum).