Both roots, plus why there are that many
Solve ax² + bx + c = 0 and see the discriminant, which tells you whether the equation has two roots, one, or none that are real.
Every quadratic follows the same formula, and the part under the square root — the discriminant — decides the shape of the answer before you finish calculating.
Positive discriminant means two real roots and a parabola crossing the axis twice. Zero means it just touches. Negative means it never reaches the axis at all.
What this calculator shows
- Both roots of the equation, real or complex
- The discriminant b² − 4ac
- A clear signal when the roots are complex rather than real
What to keep in mind
- Requires a to be non-zero; otherwise the equation is linear, not quadratic.
- Complex roots are reported in a ± bi form.
FAQs
What does the discriminant actually tell me?
How many times the curve meets the x-axis. Above zero: twice. Exactly zero: once, at the vertex. Below zero: never.
What happens if a is zero?
It stops being quadratic. bx + c = 0 is a straight line with a single root at −c/b.
Are complex roots useful or just an error?
Genuinely useful. Oscillating systems in electronics and control engineering are described precisely by complex roots.
Can I always factor a quadratic instead?
Only when the roots are rational. The formula works every time; factoring is a shortcut that often is not available.
Worked example
A factorisable quadratic
Input: x² − 5x + 6 = 0
Output: x = 3 and x = 2
Note: The discriminant is 1 — positive, so two distinct real roots, and small enough that the equation factors neatly into (x−2)(x−3).