Quadratic Equation Calculator
Roots and discriminant of ax² + bx + c = 0, including complex roots.
Quick answer
- Formula: x = (−b ± √(b² − 4ac)) ÷ 2a; Discriminant D = b² − 4ac
- Worked example — Discriminant (b² − 4ac): 1.00 (full example below).
Quadratic Equation Calculator
| Discriminant (b² − 4ac) | 1.00 |
| Roots | x = 2, x = 1 |
| Vertex x-coordinate (−b ÷ 2a) | 1.5000 |
x² − 3x + 2 = 0 has discriminant 9 − 8 = 1, and roots x = 2 and x = 1 — check: 4 − 6 + 2 = 0 and 1 − 3 + 2 = 0.
A quadratic equation has the form ax² + bx + c = 0, and its graph is a parabola. The discriminant, b² − 4ac, tells you before solving how many real solutions exist: positive means two, zero means one repeated root, and negative means the roots are complex numbers.
Enter the three coefficients. You get the discriminant, the roots as a written summary (real roots listed individually, or the complex pair described with i), and the x-coordinate of the parabola's vertex.
The formula
Worked example
x² − 3x + 2 = 0 has discriminant 9 − 8 = 1, and roots x = 2 and x = 1 — check: 4 − 6 + 2 = 0 and 1 − 3 + 2 = 0.
The example is pre-loaded in the result panel — press Calculate with your own numbers to replace it.
Frequently asked questions
What does a negative discriminant mean?
The parabola never crosses the x-axis, so there are no real roots. The solutions are complex numbers, written here as a pair like −1 + 2i and −1 − 2i, where i is the square root of −1.
What if a = 0?
Then the equation is not quadratic at all — it is linear (bx + c = 0) with the single solution x = −c/b. This calculator asks for a non-zero a.
How can I check a root?
Substitute it back into the original equation: if ax² + bx + c comes out to 0, the root is correct. Both roots of x² − 3x + 2 = 0 (2 and 1) pass this check, as the example shows.
Category: Math calculators
Sources & methodology
This calculator applies the formula shown on this page exactly as written, using the standard definitions for its inputs. Results are computed from the values you enter — no data is stored — and are rounded only for display.
Content last reviewed by CalcHub Editorial.
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