Quadratic Formula Calculator
Adjust the inputs below. Results update as you type.
How it works
x = (−b ± √(b²−4ac)) / 2a. Discriminant Δ = b²−4ac: if Δ > 0, two distinct real roots; Δ = 0, one repeated real root; Δ < 0, two complex conjugate roots. Alternatively use factoring (if factorable) or completing the square. Quadratics model projectile motion, economics (profit/revenue), and optimization problems. The vertex is at x = −b/2a.
Input guidance
- Check units and decimal placement before running calculations.
- For percentage and ratio tasks, define the baseline value clearly.
- Use rounded output carefully and keep full precision for intermediate steps.
The formula
For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b² − 4ac shows whether roots are real and distinct, repeated, or complex.
Worked example
For x² − 5x + 6 = 0: x = (5 ± √(25 − 24))/2 = (5 ± 1)/2, giving x = 3 or x = 2.
More examples to test
- Baseline example: solve with clean numbers first to verify formula direction.
- Edge-case example: test near-zero or boundary values to avoid interpretation errors.
How to interpret results
Use outputs as quick checks, then verify with step-by-step work for graded or high-stakes use.
When this can be inaccurate
Errors usually come from unit mismatch, wrong baseline assumptions, or rounding too early in multistep problems.
Change history
- July 2026: Quality-reviewed for publication with formula checks and explanatory copy updates.
Site-wide corrections also appear on the corrections log.
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Frequently asked questions
What is the discriminant?+
The b² − 4ac part. If positive there are two real roots, if zero one repeated root, and if negative two complex roots.
When do I use the quadratic formula?+
For any quadratic equation, especially when factoring is hard. It always finds the roots given a, b, and c.
What if a equals zero?+
Then it is not quadratic but linear (bx + c = 0), solved simply as x = −c/b.
Why do my classroom and calculator results differ?+
Differences usually come from rounding conventions, order-of-operations handling, or different baseline assumptions.
What is the safest way to validate an answer?+
Recompute with units shown and check against a second method or manual arithmetic for critical problems.