Quadratic Equation Calculator
Every Parabola Has a Story to Tell
Any equation of the form ax² + bx + c = 0 describes a parabola, and where that parabola crosses the x-axis — if it does at all — is exactly what the quadratic formula reveals. This calculator takes the three coefficients and returns every case the formula can produce: two distinct real roots, one repeated root, or a pair of complex roots when the parabola never touches the x-axis.
The Formula
The term under the square root, b² − 4ac, is the discriminant. Its sign determines which of the three cases applies: positive gives two real roots, zero gives one repeated root, and negative gives two complex conjugate roots.
Where Quadratics Appear
- Projectile motion — the height of a thrown or launched object over time follows a quadratic curve, and the roots are when it hits the ground.
- Profit and break-even analysis — revenue and cost functions that include a squared term produce quadratic profit equations.
- Engineering design — parabolic arcs in bridges, satellite dishes, and structural curves are described by quadratic equations.
- Algebra coursework — verifying factored or completed-square solutions against the formula.
| Equation | Discriminant | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | 1 (positive) | x = 3, x = 2 |
| x² − 4x + 4 = 0 | 0 | x = 2 (repeated) |
| x² + 2x + 5 = 0 | −16 (negative) | x = −1 + 2i, x = −1 − 2i |
How to Use This Calculator
- Enter Coefficient a (the coefficient of x²).
- Enter Coefficient b (the coefficient of x).
- Enter Coefficient c (the constant term).
- Select Calculate to see the discriminant and the resulting root(s).
Related Calculations
Need to simplify an expression or solve a linear equation instead? The Algebra Calculator handles those cases. For area and other parabola-adjacent geometry, see the Geometry Calculator.