RootFinder
Polynomial Roots Calculator
Quadratic roots, real or complex.
Roots of a quadratic and what the discriminant predicts
For ax squared plus bx plus c, the discriminant b squared minus 4ac determines how many real roots exist.
| Polynomial | Discriminant | Roots | Type |
|---|---|---|---|
| x2 - 3x + 2 | 1 | 2, 1 | Two real roots |
| x2 - 5x + 6 | 1 | 3, 2 | Two real roots |
| x2 - 4 | 16 | 2, -2 | Two real roots |
| 2x2 + 5x - 3 | 49 | 0.5, -3 | Two real roots |
| x2 + 2x + 1 | 0 | -1 | One repeated root |
| x2 + x + 1 | -3 | -0.5 +- 0.866i | Two complex roots |
Read the discriminant before solving and you already know the shape of the answer: positive gives two distinct real roots where the curve crosses the axis, zero gives a single repeated root where it just touches, and negative gives a conjugate pair of complex roots where it never touches at all. A discriminant that is a perfect square, as in the first four rows, also tells you the roots are rational and the polynomial factors neatly over the integers. The sum of the roots is always -b/a and their product c/a, which is a fast way to check.
Where the curve crosses zero
The roots of a polynomial are the x-values where it equals zero — graphically, where the curve crosses the x-axis. A quadratic can cross twice, touch once, or never cross at all (complex roots).
Why it matters beyond algebra class
Finding roots is how you solve for break-even points, the moment a projectile hits the ground, or any 'when does this equal zero' question in applied math.
Frequently asked questions
A ball is thrown upward: h = −5t² + 20t + 3. When does it hit the ground?
Set h = 0: −5t² + 20t + 3 = 0. Using the quadratic formula with a = −5, b = 20, c = 3: discriminant = 400 + 60 = 460. t = (−20 ± √460) / (−10). The positive root ≈ 4.14 seconds is when the ball lands.
How is this calculator different from the equation solver?
Both solve quadratic equations, but this tool emphasizes the roots as geometric features — where the polynomial crosses zero. The equation solver focuses on the algebraic steps of the quadratic formula itself. Use whichever framing matches your problem.
What does the discriminant tell me without solving the whole equation?
The sign of b² − 4ac tells you everything about root count: positive → 2 real roots, zero → 1 repeated root, negative → 2 complex conjugate roots. It's a quick diagnostic before doing the full computation.
Can this solver handle cubics or higher-degree polynomials?
This tool handles quadratics (degree 2). For cubics and quartics, there are general formulas but they're much more complex. For degree 5 and above, no general algebraic formula exists (proven by Abel–Ruffini). Numerical methods are needed for those.
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Last updated: September 6, 2026