Calculator Form
Enter coefficients for ax² + bx + c = 0. The calculator uses a responsive 3-column, 2-column, and 1-column layout.
Plotly Graph
The graph helps you see the parabola, vertex, and real intercepts when they exist.
Formula Used
Standard Form
ax² + bx + c = 0
Quadratic Formula
x = (-b ± √(b² - 4ac)) / (2a)
Discriminant
D = b² - 4ac
Vertex
xₚ = -b / (2a)
yₚ = f(xₚ)
A positive discriminant gives two real roots. Zero gives one repeated root. A negative discriminant gives complex roots.
The sign of a decides whether the parabola opens upward or downward.
How to Use This Calculator
- Enter the values of coefficients a, b, and c.
- Choose the decimal precision you want in the output.
- Set graph minimum and maximum x-values if you want a custom graph view.
- Click Solve Equation to calculate roots and related properties.
- Review the result block above the form for roots, discriminant, vertex, and factor form.
- Use the graph to inspect the parabola shape and intercept behavior.
- Download the result summary as CSV or PDF for records or sharing.
Example Data Table
| Equation | a | b | c | Discriminant | Roots | Nature |
|---|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 | 1 | 2, 3 | Two distinct real roots |
| x² + 4x + 4 = 0 | 1 | 4 | 4 | 0 | -2 | One repeated real root |
| x² + 2x + 5 = 0 | 1 | 2 | 5 | -16 | -1 ± 2i | Complex conjugate roots |
| 2x² - 3x - 2 = 0 | 2 | -3 | -2 | 25 | 2, -0.5 | Two distinct real roots |
FAQs
1) What happens if a = 0?
Then the relation is not quadratic. This calculator still explains the linear or constant case so you can spot input mistakes immediately.
2) Why is the discriminant important?
It reveals the root type. Positive gives two real roots, zero gives one repeated root, and negative gives a complex conjugate pair.
3) Why does the graph matter?
The graph shows the parabola shape, turning point, and intercepts. It is useful for checking whether numerical results match the visual behavior.
4) Can this calculator show complex roots?
Yes. When the discriminant is negative, the calculator displays the two complex conjugate roots in a readable form.
5) What is the vertex of a quadratic?
The vertex is the turning point of the parabola. It gives the minimum value when the curve opens upward and the maximum value when it opens downward.
6) Are the displayed values exact?
Displayed decimals are rounded to your chosen precision. Internal calculations use floating-point math, so irrational roots and repeating decimals are approximated.
7) Can I export the calculated result?
Yes. The result block includes CSV and PDF download buttons so you can save, print, or share the solution summary.
8) Which equation form is best for graphing?
Vertex form is often best for graphing insight, while standard form is best for coefficient entry and discriminant-based solving.