How Many Zeros in a Quadratic Polynomial?
A quadratic polynomial has exactly 2 zeros in the complex number system, counting multiplicity. A quadratic has the form f(x) = ax2 + bx + c (a ≠ 0), and the Fundamental Theorem of Algebra guarantees exactly 2 roots. The nature of those zeros depends on the discriminant (b2 − 4ac): if positive, there are 2 distinct real zeros; if zero, there is 1 repeated real zero (counted twice); if negative, there are 2 complex conjugate zeros with no real solutions. In all cases, the total is always exactly 2. Related: Polynomial function zeros.
A quadratic polynomial has
2
zeros
- Written Form
- f(x) = ax² + bx + c (where a ≠ 0)
- Scientific
- Degree 2
How Many Zeros Does a Quadratic Polynomial Have?
A quadratic polynomial always has exactly 2 zeros, but the type varies based on the discriminant Δ = b2 − 4ac: See also: Even degree polynomial zeros.
| 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.
How Do You Find the Zeros of a Quadratic Polynomial?
Three main methods work for any quadratic ax2 + bx + c: Related: Zeros in a polynomial with imaginary zeros.
- Factoring: Rewrite as (x − r)(x − s) = 0, giving zeros x = r and x = s. Works cleanly when the roots are integers or simple fractions.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a — works for any quadratic, including those with complex or irrational roots.
- Completing the square: Rewrite ax2 + bx + c in the form a(x − h)² + k, then solve (x − h)² = −k/a. Useful for understanding vertex form.
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. The polynomial has 2 zeros, consistent with its degree.
Can a Quadratic Polynomial Have More Than 2 Zeros?
No. A non-zero polynomial of degree n can have at most n zeros. A quadratic is degree 2, so it has at most 2 zeros — and exactly 2 when complex roots are counted. If a quadratic appeared to have 3 or more zeros, it would contradict the Fundamental Theorem of Algebra. The only polynomial with infinitely many zeros is the zero polynomial, f(x) = 0, which is not a true quadratic (or any degree).