Solves any quadratic equation of the form ax² + bx + c = 0 for its roots, using the quadratic formula.
How it works
The discriminant (b² − 4ac) is computed first to check how many real roots exist, then the quadratic formula gives the root or roots directly.
What this does not include
This does not include converting the equation to vertex form to reveal the parabola’s vertex — for that, use this site’s vertex form calculator instead.
How to use this calculator
- Enter the a, b, and c coefficients from your equation.
A worked example
For x² − 5x + 6 = 0 (a=1, b=−5, c=6): discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1. Roots = (5 ± 1) ÷ 2 = 3 and 2.
For x² + 2x + 5 = 0 (a=1, b=2, c=5): discriminant = 4 − 20 = −16 — no real roots.
What the variables mean
| Variable | Meaning |
|---|---|
| a, b, c | Coefficients from ax² + bx + c = 0 |
| Discriminant (b² − 4ac) | Predicts the number and type of roots before solving |
Edge cases worth knowing
A discriminant of exactly zero gives one repeated root, not two distinct ones — the parabola just touches the x-axis at a single point.
A of zero isn’t a quadratic equation at all. Without the x² term the equation is linear, and the quadratic formula doesn’t apply — this calculator requires a nonzero a.
Frequently asked questions
What does the discriminant tell you?
Its sign predicts the type of roots: positive means two distinct real roots, zero means exactly one repeated real root, and negative means no real roots (only complex ones).
Why can’t a be zero?
If a is zero, the a x² term disappears and the equation becomes linear (bx + c = 0), not quadratic — the quadratic formula no longer applies.
What does “no real roots” mean graphically?
The parabola never crosses the x-axis — it stays entirely above or entirely below it.
