Quadratic Equation Solver Guide: Formula & Worked Example
Short answer
Solve ax² + bx + c = 0, including complex roots. In the fixed example below, roots is x₁ = 2, x₂ = 1. Your answer depends on the inputs and assumptions you choose.
Formula
x = (−b ± √(b² − 4ac)) / 2a
Use the relationship above with consistent units. The inputs are a, b, c. Use the input scale shown beside each field and keep time periods and units consistent. Only round the final result for display.
Worked example
- a
- 1
- b
- -3
- c
- 2
- Roots: x₁ = 2, x₂ = 1
- Discriminant: 1
- Vertex: (1.5, -0.25)
Understanding the result
Solve ax² + bx + c = 0, including complex roots. Start with the base example, then replace each value with the figure that matches your situation.
a
The example uses 1 for a. This is an illustration, not a recommended value. Keep the unit and time period shown by the field.
b
The example uses -3 for b. This is an illustration, not a recommended value. Keep the unit and time period shown by the field.
c
The example uses 2 for c. This is an illustration, not a recommended value. Keep the unit and time period shown by the field.
What the comparison means
The comparison changes only a, keeping every other example input fixed. This isolates that input's effect on the primary result. It does not predict future outcomes.
Check units, date boundaries and rounding before using the answer in a report or a practical measurement.
Comparison
| Scenario | Input | Result |
|---|---|---|
| Lower input | a: 0.8 | Roots: x₁ = 2.882782, x₂ = 0.867218 |
| Base example | a: 1 | Roots: x₁ = 2, x₂ = 1 |
| Higher input | a: 1.2 | Complex roots: 1.25 ± 0.322749i |
Common mistakes
- Using the wrong units for a.
- Comparing results while changing more than one input at a time.
- Check units, date boundaries and rounding before using the answer in a report or a practical measurement.
Checklist
- Confirm a.
- Confirm b.
- Confirm c.
- Check the result against your source or provider.
Calculators Percentage Editorial Team
Calculator guides from the Calculators Percentage team. These introductory examples are educational, not professional advice.
Last reviewed:
Check units, date boundaries and rounding before using the answer in a report or a practical measurement.