x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}Variables
- a: coefficient of x² (a ≠ 0)
- b: coefficient of x
- c: constant term
- x: solution or root
How to use this formula
Finds the roots of a quadratic equation ax² + bx + c = 0 when a is nonzero.
Important notes
- The discriminant b² − 4ac determines the number and type of real roots.
- Use the ± sign to calculate both possible roots.
Quick example
For x² − 5x + 6 = 0, the roots are x = 2 and x = 3.
Applicability, worked calculation, and verification
Domain and applicability
Use for equations in the form ax² + bx + c = 0 with a ≠ 0; complex roots are allowed when the discriminant is negative.
Units
- a, b, c, and x use compatible problem-specific units
Assumptions and domain checks
- The discriminant b² − 4ac determines the number and type of real roots.
- For the Quadratic Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.
- The plus-minus sign represents separate branches; evaluate and verify both when both are admissible.
Do not use this formula when
- Do not use Quadratic Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Quadratic Formula, check zero, negative, and extreme input values before relying on the result.
- When using Quadratic Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Solve x² − 5x + 6 = 0. Here a = 1, b = −5, and c = 6. The discriminant is 1, so the two branches give x = 3 and x = 2.
- Identify the coefficients from standard form: a = 1, b = −5, and c = 6.
- Compute the discriminant: b² − 4ac = 25 − 24 = 1.
- Substitute into the formula: x = (5 ± √1) / 2.
- Evaluate both branches to obtain x = 3 and x = 2.
Independent verification
Substituting 3 gives 9 − 15 + 6 = 0, and substituting 2 gives 4 − 10 + 6 = 0, so both roots satisfy the original equation.
Common mistakes
- Do not collapse the ± in Quadratic Formula into one value; calculate each admissible branch and check it in the original problem.
- For the Quadratic Formula, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Quadratic Formula 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
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. - 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 Formula.
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 Formula used for?
Finds the roots of a quadratic equation ax² + bx + c = 0 when a is nonzero.
Can I copy this formula as LaTeX?
Yes. Copy x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} 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.