Quadratic Formula Tool
Enter quadratic coefficients to find real or complex roots, the discriminant, vertex, axis of symmetry and intercept. Explore an interactive parabola, review substitution steps and equation forms, adjust numerical precision, and export results or graphs.
| x | f(x) | Position |
|---|
Table of Contents
Highlights
- Find two real, repeated or complex roots
- Explore an interactive parabola with selectable markers
- See discriminant, vertex, symmetry axis and intercept
- Review formula steps, equation forms and value table
- Export results as CSV or JSON and graphs as SVG or PNG
How Quadratic Formula Tool Works?
A quadratic equation has the standard form ax² + bx + c = 0, where a, b and c are real coefficients and a ≠ 0. The two roots follow from the quadratic formula:
The discriminant, Δ = b² − 4ac, determines whether the roots are distinct, repeated or complex.
| Discriminant | Roots | Graph |
|---|---|---|
| Δ > 0 | Two distinct real roots | Crosses the x-axis twice |
| Δ = 0 | One repeated real root | Touches the x-axis at its vertex |
| Δ < 0 | Two complex conjugate roots | Does not cross the x-axis |
The vertex occurs at h = −b/(2a) and k = ah² + bh + c; the symmetry axis is x = h, and the y-intercept is (0, c). A positive a makes the parabola open upwards and a negative a makes it open downwards. The tool also displays standard, vertex and real factored forms where available.
For example, entering a = 1, b = −3, c = 2 gives Δ = 1, roots x = 1 and x = 2, and vertex (1.5, −0.25). You can change decimal places, number notation and the graph range, show or hide roots, vertex and grid, inspect graph coordinates, and view a nine-row x–f(x) value table.
Practical Use Cases
Students: Check solutions to exercises such as x² − 3x + 2 = 0, compare the discriminant with the graph, and review the quadratic formula substitution step by step.
Teachers and tutors: Demonstrate how changing a, b or c moves the vertex, changes the opening direction and alters where a parabola meets the x-axis.
Engineers: Inspect real roots of simple quadratic models and use the vertex to locate a modelled minimum or maximum, while checking that the model is suitable for the problem.
Data analysts: Explore a fitted quadratic trend, identify its turning point and inspect predicted function values across the displayed x-range.
Graphic designers and animators: Explore parabola shape and symmetry for curved trajectories or layouts, then export the plotted curve as SVG or PNG for reference.
Accuracy and Limitations
Calculations use JavaScript double-precision floating-point arithmetic, so displayed roots, coordinates and discriminants are approximations rather than exact symbolic fractions or radicals. Decimal places (2–12) and automatic, decimal or scientific notation change the display, not the underlying precision. The tool requires finite real coefficients, rejects a = 0 and coefficient magnitudes above 10¹⁰⁰, and treats sufficiently small scaled discriminants as zero using a relative tolerance of 10⁻¹⁴. The graph samples 241 points across the selected range, and its value table contains nine samples; extreme coefficient combinations may produce unstable or unplottable values. Complex roots are listed numerically but are not plotted as real x-axis intercepts, and factored form is not provided over the real numbers when roots are complex.
