\Delta=b^2-4acVariables
- a, b, c: coefficients of ax²+bx+c
- Δ: discriminant
How to use this formula
Determines the number and type of roots of a quadratic equation.
Important notes
- Δ>0 gives two distinct real roots; Δ=0 gives one repeated real root.
Quick example
For x²−4x+4, Δ=0.
Applicability, worked calculation, and verification
Domain and applicability
Apply Quadratic Discriminant only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- Δ>0 gives two distinct real roots; Δ=0 gives one repeated real root.
- For the Quadratic Discriminant, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Quadratic Discriminant when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Quadratic Discriminant, check zero, negative, and extreme input values before relying on the result.
- When using Quadratic Discriminant, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \Delta=b^2-4ac for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For x²-4x+4=0, Δ=(-4)²-4(1)(4)=16-16=0.
- Identify a=1, b=-4, and c=4.
- Square b and compute 4ac.
- Subtract to obtain Δ=0, indicating one repeated real root.
Independent verification
Factoring gives (x-2)²=0, confirming the repeated root x=2.
Common mistakes
- Before substituting values into Quadratic Discriminant, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Quadratic Discriminant, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Quadratic Discriminant in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\Delta=b^2-4ac. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Quadratic Discriminant.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Quadratic Discriminant used for?
Determines the number and type of roots of a quadratic equation.
Can I copy this formula as LaTeX?
Yes. Copy \Delta=b^2-4ac or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.