Quadratic Formula Calculator
Roots, discriminant, vertex and axis of symmetry.
For ax² + bx + c = 0. a cannot be zero.
The formula
The discriminant decides everything
The quantity under the root, b² − 4ac, is the discriminant. Its sign tells you what kind of roots to expect before you finish the calculation.
| Discriminant | Roots | Graph |
|---|---|---|
| Positive | Two distinct real roots | Crosses the x-axis twice |
| Zero | One repeated real root | Touches the x-axis once |
| Negative | Two complex conjugates | Never meets the x-axis |
A negative discriminant is not an error. The roots exist, they are just complex, and the calculator reports them in a ± bi form.
Vertex and axis of symmetry
The parabola is symmetric about x = −b/2a, which is the average of the two roots. Substituting that back gives the vertex — the minimum when a is positive, the maximum when a is negative.
Sign errors, which are the usual problem
In x² − 3x + 2, b is −3, not 3. Then b² is (−3)² = 9, not −9. Squaring a negative gives a positive, and writing −3² without brackets means −(3²). The calculator prints the substitution with brackets for exactly this reason.
When factoring is faster
If the discriminant is a perfect square the roots are rational and factoring is usually quicker by hand. The formula always works; factoring only sometimes does.