algebra
The Discriminant Explained
January 11, 2026
4 min read
PanMaths Team
⚡ THE DISCRIMINANT
D = b² - 4ac
D > 0: Two real roots | D = 0: One repeated root | D < 0: Complex roots
The Three Cases
D > 0
Two Distinct Real Roots
Parabola crosses x-axis at TWO points
x² - 5x + 6 = 0 → D = 25 - 24 = 1 → x = 2, 3D = 0
One Repeated Real Root
Parabola TOUCHES x-axis at ONE point
x² - 4x + 4 = 0 → D = 16 - 16 = 0 → x = 2 (double)D < 0
Two Complex Roots
Parabola NEVER crosses x-axis
x² + 2x + 5 = 0 → D = 4 - 20 = -16 → x = -1 ± 2i💡Why Does This Work?
The quadratic formula has √(b² - 4ac) in it. If this is positive, you can take the square root and get two real numbers. If zero, you get one answer. If negative, you get imaginary numbers (i).
Calculate the Discriminant
Enter your coefficients and find out what type of roots you'll get:
x² +x += 0
Quick Reference
| Discriminant | Roots | Type |
|---|---|---|
| D > 0 | 2 | Real, different |
| D = 0 | 1 | Real, repeated |
| D < 0 | 0 real | Complex |