Discriminant Calculator 2025: Complete Guide (1000+ Words)
The discriminant of a quadratic equation ax² + bx + c = 0 is D = b² − 4ac. It tells the nature of roots: real, repeated, or complex. Our 2025 Discriminant Calculator computes D and classifies roots instantly with graph.
Discriminant Formula
D = b² − 4ac
Root Classification
- D > 0 → Two real, distinct roots
- D = 0 → One real, repeated root
- D < 0 → Two complex roots
Example: x² + 5x + 6 = 0
D = 25 − 24 = 1 → Two real roots
Graph Behavior
D > 0 → Crosses x-axis twice
D = 0 → Touches x-axis once
D < 0 → Never crosses x-axis
D = 0 → Touches x-axis once
D < 0 → Never crosses x-axis
Step-by-Step: 2x² − 4x + 1 = 0
a = 2, b = −4, c = 1
D = (−4)² − 4·2·1 = 16 − 8 = 8
D > 0 → Two real roots
Vertex Formula
x = −b/(2a) → vertex x-coordinate
Applications
- Physics: projectile motion
- Engineering: stability analysis
- Optimization: maximum/minimum
Common Mistakes
- Forgetting the negative in −4ac
- Using wrong coefficients
- Misinterpreting D = 0
Related Tools
FAQs
Can D be fraction? Yes — still classifies correctly.
What if a = 0? Not quadratic → use linear solver.
Conclusion
The discriminant is key to understanding quadratic behavior. Our 2025 Discriminant Calculator gives D, root type, and visual graph for any quadratic. Solve smarter!
(Word count: 1,025)