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Formula used
The quadratic formula solves every valid quadratic equation. The discriminant identifies the root type. The vertex formulas describe the parabola.
How to use this calculator
- Select the mode matching your available information.
- Enter coefficients, an equation, roots, or geometric data.
- Choose the required decimal precision and answer format.
- Press Calculate to solve and analyze the quadratic.
- Review roots, forms, properties, graph, and verification results.
- Copy, print, or export the completed calculation.
Worked examples
| Equation | Discriminant | Root type | Solutions |
|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | Two real roots | 2 and 3 |
| x² - 4x + 4 = 0 | 0 | Repeated root | 2 |
| x² + 4x + 8 = 0 | -16 | Complex roots | -2 ± 2i |
| 2x² - 8 = 0 | 64 | Two real roots | -2 and 2 |
| 3x² + 7x - 6 = 0 | 121 | Two real roots | 2/3 and -3 |
Understanding quadratic equations
A quadratic equation has degree two. Its leading coefficient cannot equal zero. Its graph forms a parabola.
Coefficient a controls direction and width. Coefficient b influences horizontal placement. Coefficient c gives the y-intercept.
Positive discriminants produce two real roots. Zero produces one repeated root. Negative values produce complex conjugates.
Frequently asked questions
1. What is a quadratic equation?
It is an equation with highest exponent two. Its standard form is ax² + bx + c = 0. The value of a cannot be zero.
2. What does the discriminant show?
The discriminant is b² - 4ac. It predicts the number and type of roots. It also helps identify factorability.
3. Can this calculator solve complex roots?
Yes. Negative discriminants produce complex conjugate roots. Results appear in exact and decimal complex forms.
4. Can I enter fractions?
Yes. Enter values like 1/2 or -3/4. The denominator must never equal zero.
5. Why must coefficient a be nonzero?
A zero value removes the squared term. The equation then becomes linear. It is no longer quadratic.
6. How is the vertex calculated?
The x-coordinate is -b divided by 2a. Substitute that value into the function. The result gives the y-coordinate.
7. What is vertex form?
Vertex form is a(x - h)² + k. The vertex is located at (h, k). This form highlights transformations clearly.
8. Can the calculator verify proposed roots?
Yes. Verification substitutes each proposed value into the equation. A near-zero residual confirms the root.
9. Can I export my calculation?
Yes. Results can be copied, printed, or downloaded. CSV and PDF export buttons are included.