Quadratic Formula Calculator
Solve quadratic equations of the form ax² + bx + c = 0. Find real and complex roots instantly.
Equation Coefficients
Solution Results
0
b² - 4ac
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nature of roots
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first solution
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second solution
💡 Tip: If the discriminant is positive, there are two real roots. If zero, one real root. If negative, two complex roots.
The Quadratic Formula
The quadratic formula is used to find the roots (solutions) of any quadratic equation in the form ax² + bx + c = 0:
x = (-b ± √(b² - 4ac)) / 2a
The Discriminant
The discriminant (Δ = b² - 4ac) determines the nature of the roots:
| Discriminant | Root Type | Graph Behavior |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One real root (repeated) | Parabola touches x-axis once |
| Δ < 0 | Two complex roots | Parabola does not cross x-axis |
Step-by-Step Example
Solve: x² - 5x + 6 = 0 (a=1, b=-5, c=6)
- Calculate discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
- Since Δ > 0: Two real roots exist.
- Apply formula: x = (5 ± √1) / 2
- Root 1: (5 + 1) / 2 = 3
- Root 2: (5 - 1) / 2 = 2
Solutions: x = 3 and x = 2
Real-World Applications
- Physics: Projectile motion, calculating trajectories.
- Engineering: Structural analysis, electrical circuits.
- Economics: Profit maximization, break-even analysis.
- Architecture: Designing arches and parabolic structures.
References
- Khan Academy. "Quadratic Formula." 2026.
- Purplemath. "The Quadratic Formula." 2026.