Quadratic Equation Roots Calculator
Use our free online Quadratic Formula Calculator to perform fast, accurate calculations with instant formulas, step-by-step arithmetic, and verified reference benchmarks.
Quadratic Formula Calculator
Instant Live Computation • Zero Waiting
Adjust inputs on the left for instantaneous live computation.
How to Calculate: The Quadratic Formula Calculator Formula
Calculates roots using the discriminant Δ = b² - 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, one real root. If Δ < 0, two complex roots.
Step-by-Step Calculation Guide
- Identify coefficients a, b, and c in standard form ax² + bx + c = 0.
- Input coefficients into the calculator.
- Evaluate the discriminant (b² - 4ac).
- Review root solutions x₁ and x₂ alongside parabola vertex coordinates (-b/2a).
Practical Quadratic Formula Calculator Examples
Equation: x² - 5x + 6 = 0 (a=1, b=-5, c=6).
Equation: x² - 4x + 4 = 0 (a=1, b=-4, c=4).
Discriminant (Δ = b² - 4ac) Root Classification
Official reference values and benchmark classifications based on standard institutional guidelines.
| Discriminant Value | Nature of Roots | Parabola Graph Intercepts | Factoring Status |
|---|---|---|---|
| Δ > 0 (Positive) | Two distinct real roots | Crosses x-axis at two points | Solvable over real numbers |
| Δ = 0 (Zero) | One repeated real root | Touches x-axis at exactly one point (vertex) | Perfect square trinomial |
| Δ < 0 (Negative) | Two complex conjugate roots (a ± bi) | Does not intersect x-axis | Solvable over complex numbers |
Frequently Asked Questions about Quadratic Formula Calculator
What is standard form of a quadratic equation?
Standard form is ax² + bx + c = 0, where a, b, and c are known constants and a cannot equal zero.
What is the vertex of a parabola?
The vertex is the extreme peak (maximum) or valley (minimum) of the parabola, located at x = -b / (2a).
What happens if coefficient 'a' equals zero?
If a = 0, the equation reduces to bx + c = 0, which is a linear equation rather than a quadratic.
How do complex roots work?
When the discriminant is negative, taking the square root requires the imaginary unit i (where i = √-1), producing conjugate pairs like 2 + 3i and 2 - 3i.