Tools HubQuadratic Equation
Solve ax² + bx + c = 0 with real or complex roots, the discriminant and substitution into the quadratic formula.
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Key Formula & Conditions
Learning modelEnter a, b and c from ax² + bx + c = 0 to find the discriminant D = b² − 4ac and the roots. Positive D gives two distinct real roots, zero gives a repeated root, and negative D gives two complex roots.
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Result
Worked steps
Approximate results are displayed to at most 10 significant digits. Measurement significant figures and uncertainty are not inferred.
Formula and example
Enter a, b and c from ax² + bx + c = 0 to find the discriminant D = b² − 4ac and the roots. Positive D gives two distinct real roots, zero gives a repeated root, and negative D gives two complex roots.
x = (−b ± √(b² − 4ac)) / (2a)
a = 1, b = −5, c = 6 → D = 1, x = 2, 3
- Two real rootsx² − 5x + 6 = 0 → x = 2, 3
- Repeated rootx² − 6x + 9 = 0 → x = 3 (repeated root)
- Complex rootsx² + 4 = 0 → x = 0 ± 2i
Common mistakes
- Move all terms to one side before entering coefficients. For x² = 5x − 6, use a=1, b=−5, c=6.
- Use b=0 if the linear term is absent. An equation with a=0 is not quadratic.
Check the assumptions
Requires a ≠ 0. A negative discriminant produces complex roots. Floating-point rounding can affect coefficients extremely close to a repeated root.
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OpenStax · College Algebra 2e — Quadratic Equations ↗Done
Quadratic Equation
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Verification scope and citation information
Use worked examples within their stated inputs, units and assumptions. Tests cover specific cases at a specific time; they do not certify every input, expert review or standards compliance. Check the references and tool-specific verification notes, and enter your own access date when citing.
@misc{tools_hub_quadratic_equation_2026,
author = {{Tools Hub}},
title = {Quadratic Equation},
year = {2026},
url = {https://the-tools-hub.net/en/tools/quadratic-equation/},
note = {Accessed: YYYY-MM-DD}
}How to use
- 1
Enter values and units
Use the example or enter your own conditions.
- 2
Calculate
Check the result and each calculation step.
- 3
Check assumptions
Review applicability and display precision, then copy inputs and results if needed.
Features
- Worked steps and unit conversions
- No account; calculation stays in your browser
- Copy inputs and results together
Use cases
Check your working
Compare formula substitutions and units with your own calculations.
Details
Requires a ≠ 0. A negative discriminant produces complex roots. Floating-point rounding can affect coefficients extremely close to a repeated root.
FAQ
Can this calculator find complex roots?
Yes. It uses the imaginary unit i. For example, x²+4=0 has roots x=±2i.
How can I check a root?
Substitute the root into the original equation. With x=2 in x²−5x+6, the left side is 4−10+6=0.
Are results exact?
Results use floating-point approximations, displayed to at most 10 significant digits. Measurement significant figures and uncertainty are not inferred.
Are inputs saved?
Inputs and results are not sent to a server or saved. They disappear when you leave the page.
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Verified: Known values (x²−5x+6=0 → x=2, 3), repeated and complex roots, the a = 0 case and the working display are covered by browser tests