Quadratic equation solver
Solve any equation of the form ax² + bx + c = 0 and see the parts that matter: the discriminant, both roots, the vertex and the axis of symmetry. Algebra students and teachers use it to check work.
- Runs in your browser
- Free, no account needed
- Copy or download the result
Result
Example
a (x² coefficient): 1 b (x coefficient): -3 c (constant): -4
Equation: x² - 3x - 4 = 0 Discriminant (b² - 4ac): 25 Two real roots (exact rational numbers): x1 = 4 x2 = -1 Vertex: (3/2, -25/4) ≈ (1.5, -6.25) Axis of symmetry: x = 3/2 The parabola opens upward, so the vertex is a minimum. Decimals are rounded to 12 significant digits.
How to use the quadratic equation solver
- Type the coefficients in a (x² coefficient), b (x coefficient) and c (constant).
- Press Run to see the discriminant, roots, vertex and axis of symmetry.
- Press Copy result or Download to keep the solution.
- Press Reset to go back to the example x² - 3x - 4 = 0.
How it works
Roots come from the quadratic formula x = (-b ± √(b² - 4ac)) / (2a). The discriminant b² - 4ac decides whether there are two real roots, one repeated root or two complex roots.
Coefficients are scaled to whole numbers first, so rational roots are shown as exact fractions. Other roots get an exact form with a simplified square root, such as (2 ± √6) / 2, plus decimals.
The vertex is (-b / 2a, c - b² / 4a) and the axis of symmetry is x = -b / 2a.
If a is 0, the equation is solved as linear. Coefficients may have up to 6 decimal places and a size up to 1e9.
Everything runs in your browser. Your input is not uploaded or stored.
Questions and answers
What does the discriminant tell you?
If b² - 4ac is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the roots are complex.
Can this solver give complex roots?
Yes. When the discriminant is negative it shows the roots as p ± qi, in exact form such as (-3 ± i√23) / 4 and as decimals.
What happens if a = 0?
The equation is no longer quadratic. The tool solves bx + c = 0 instead, or tells you if there is no solution or every x works.