How Many Zeros in a Quadratic Polynomial?
How many zeros in a quadratic polynomial? A quadratic polynomial has exactly 2 zeros in the complex number system, counting multiplicity — never more, never fewer. A quadratic has the form f(x) = ax2 + bx + c (a ≠ 0), and the FTA guarantees exactly 2 roots. The nature of those zeros depends on the discriminant (b2 − 4ac). Related: Zeros in a constant polynomial.
A quadratic polynomial has
2
zeros
- Written Form
- f(x) = ax² + bx + c (where a ≠ 0)
- Scientific
- Degree 2
The Discriminant and a Quadratic's Zero Count
The direct answer: always exactly 2, but the type varies based on the discriminant Δ = b2 − 4ac:
| Discriminant | Real zeros | Complex zeros | Example |
|---|---|---|---|
| Δ > 0 | 2 distinct real | 0 | x² − 5x + 6 → x = 2, x = 3 |
| Δ = 0 | 1 repeated real | 0 | x² − 6x + 9 → x = 3 (×2) |
| Δ < 0 | 0 | 2 complex conjugates | x² + 4 → x = ±2i |
A quadratic can have 0, 1, or 2 real zeros — but always exactly 2 zeros total when complex roots are included. Related: Sextic polynomial zeros.
How Do You Find the Zeros of a Quadratic Polynomial?
- Factoring
- Rewrite as (x − r)(x − s) = 0 — works when roots are integers or simple fractions
- Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a— works for any quadratic- Completing the square
- Rewrite as a(x − h)² + k, then solve (x − h)² = −k/a
For example, to find the zeros of f(x) = 293x2 − 293x: factor as 293x(x − 1) = 0, giving zeros at x = 0 and x = 1.
Can a Quadratic Polynomial Have More Than 2 Zeros?
- No — a non-zero polynomial of degree n has at most n zeros
- A quadratic is degree 2, so it has at most 2 zeros, exactly 2 when complex roots are counted
- The only polynomial with infinitely many zeros is the zero polynomial itself
Zeros in a Quadratic Polynomial: FAQ
Can a quadratic have 3 zeros?
Quick answer
No — that would contradict the Fundamental Theorem of Algebra; a degree-2 polynomial has at most 2.
What determines whether zeros are real or complex?
Quick answer
The discriminant b² − 4ac: positive gives real zeros, negative gives complex conjugates. Related: How many zeros in a polynomial function.
Can a quadratic have exactly 1 zero?
Yes, in the sense of one repeated (double) root when the discriminant is zero — technically still 2 zeros counted with multiplicity.
What's an example of a quadratic with no real zeros?
f(x) = x² + 4, whose zeros are the complex pair x = ±2i.
Do all quadratics have a real x-intercept?
No — only when the discriminant is non-negative; otherwise the graph never touches the x-axis.