Quadratic Equation Solver Calculator
Solve ax² + bx + c = 0. Gives both roots (real or complex), the discriminant and the vertex of the parabola, with the working.
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Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
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About this calculator
This page calculates root x₁, real part, root x₁, imaginary part, root x₂, real part and 4 more results from coefficient a, coefficient b and constant c. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
Quadratics appear when a circuit or a motor equation is squared away: the resonance and damping of a series RLC circuit, the voltage that gives a stated power through a source resistance, and the cable size that meets a limit when current depends on size. Check a root by substituting it back. For complex numbers use the complex number calculator.
Cross-check: 25 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 25 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| Coefficient a (of x²) | 1 |
| Coefficient b (of x) | -5 |
| Constant c | 6 |
- D = b² − 4ac = -5² − 4 × 1 × 6 = 1
- D > 0: two real roots, x = (−b ± √D) / 2a, √D = 1
- x₁ = 3 x₂ = 2
- Vertex: x = −b / 2a = 2.5, y = -0.25
Assumptions
- The coefficients are plain real numbers
- x₁ is the root taken with +√D and x₂ with −√D, so for a negative a the larger root is x₂
- For real roots the smaller-magnitude root is found from x₁ × x₂ = c / a, which avoids losing digits to cancellation
- A negative discriminant gives a complex-conjugate pair; the imaginary parts are shown with their signs
- The vertex is the turning point of y = ax² + bx + c
Quadratics appear when a circuit or a motor equation is squared away: the resonance and damping of a series RLC circuit, the voltage that gives a stated power through a source resistance, and the cable size that meets a limit when current depends on size. Check a root by substituting it back. For complex numbers use the complex number calculator.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| Coefficient a (of x²) | (number) | any number | Cannot be zero; if it is, the equation is linear and has one root, −c / b. |
| Coefficient b (of x) | (number) | any number | |
| Constant c | (number) | any number |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| Root x₁, real part (with +√D) | (number) | Shown to 8 decimal places | |
| Root x₁, imaginary part | (number) | Shown to 8 decimal places | |
| Root x₂, real part (with −√D) | (number) | Shown to 8 decimal places | |
| Root x₂, imaginary part | (number) | Shown to 8 decimal places | |
| Discriminant D = b² − 4ac | (number) | Shown to 8 decimal places | |
| Vertex x (−b / 2a) | (number) | Shown to 8 decimal places | |
| Vertex y (value at the vertex) | (number) | Shown to 8 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with coefficient a (of x²)
| Coefficient a (of x²) | Root x₁, real part | Root x₁, imaginary part | Root x₂, real part |
|---|---|---|---|
| 0.1 | 48.76972865 | 0 | 1.23027135 |
| 0.25 | 18.71779789 | 0 | 1.28220211 |
| 0.5 | 8.60555128 | 0 | 1.39444872 |
| 1 | 3 | 0 | 2 |
| 2 | 1.25 | 1.19895788 | 1.25 |
| 4 | 0.625 | 1.05326872 | 0.625 |
| 10 | 0.25 | 0.73314391 | 0.25 |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter coefficient a (of x²), coefficient b (of x) and constant c. Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
The coefficients are plain real numbers. x₁ is the root taken with +√D and x₂ with −√D, so for a negative a the larger root is x₂. For real roots the smaller-magnitude root is found from x₁ × x₂ = c / a, which avoids losing digits to cancellation. A negative discriminant gives a complex-conjugate pair; the imaginary parts are shown with their signs. The vertex is the turning point of y = ax² + bx + c.
Which calculators should I use next?
Common next steps are Complex Numbers (Rectangular and Polar), Series RLC Impedance, Trigonometry (Sin, Cos, Tan) and Logarithm Calculator. The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 25 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Quadratic, Logarithm and Power Formulas: Formulas for the quadratic equation, logarithms and exponents with electrical examples: real and complex roots, decibels
- Complex Number Formulas: Polar and Rectangular: Complex number formulas for engineers: rectangular and polar form, conversion with atan2, addition
- Resonance Formulas: Frequency, Q, Bandwidth: Resonance formulas for a series or parallel LC circuit: resonant frequency, reactance at resonance
More Engineering Mathematics calculatorsAll engineering calculatorsQuadratic Equation Solver in Elecmetric