# Quadratic Formula Calculator

> Solve ax² + bx + c = 0: real or complex roots, discriminant, vertex and axis of symmetry.

Page: https://freecalculator.tech/tools/quadratic-formula-calculator
Last reviewed: 2026-11-03

## Formula

For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac; vertex h = −b ÷ 2a, k = a h² + b h + c.

- Source: OpenStax College Algebra 2e §2.5, Quadratic Equations.
- D > 0 gives two real roots, D = 0 one repeated root, D < 0 two complex conjugate roots.
- For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac; vertex h = −b ÷ 2a, k = a h² + b h + c.
- Inputs used: Coefficient a (x²), Coefficient b (x), Constant c.

## Inputs

- Coefficient a (x²)
- Coefficient b (x)
- Constant c

## Worked example: x² − 5x + 6 = 0

Roots 3 and 2; discriminant 1; vertex (2.5, −0.25).
1. D = 25 − 24 = 1.
2. x = (5 ± 1) ÷ 2 → 3 and 2.
3. Vertex h = 2.5, k = −0.25.

## Common questions

### What does a negative discriminant mean?

There are no real roots; the calculator shows the two complex conjugate roots.

### Why can't a be zero?

Without the x² term the equation is linear and the quadratic formula divides by zero.

### What is the vertex?

The turning point of the parabola y = ax² + bx + c.

### How do I use the Quadratic Formula Calculator?

Enter the required values for Coefficient a (x²), Coefficient b (x), Constant c. The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.

### What formula does the Quadratic Formula Calculator use?

For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ 2a. Discriminant D = b² − 4ac; vertex h = −b ÷ 2a, k = a h² + b h + c. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.

### Can the Quadratic Formula Calculator be used for exact decisions?

Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.

## Limitations and assumptions

- Real-number coefficients only.
- Displayed roots are rounded; very large or tiny coefficients can lose floating-point precision.
- Exact mathematical relationships, rounded for display. Check units and rounding before using a result in graded, engineering or financial work.
- The result depends on the values you enter for this quadratic formula calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.

## Sources

- OpenStax — https://openstax.org/books/college-algebra-2e/pages/2-5-quadratic-equations

## Cite this page

- APA: Free Calculator. (2026, November 3). Quadratic Formula Calculator. https://freecalculator.tech/tools/quadratic-formula-calculator
- MLA: "Quadratic Formula Calculator." Free Calculator, 3 Nov. 2026, https://freecalculator.tech/tools/quadratic-formula-calculator.
- Link: Quadratic Formula Calculator — Free Calculator, https://freecalculator.tech/tools/quadratic-formula-calculator