Discriminant Roots Chart- Determining Root Types Made Easy
What the Discriminant Actually Does
The discriminant is the part under the square root in the quadratic formula. It tells you what kind of roots your equation has before you solve anything.
For a quadratic equation ax² + bx + c = 0, the discriminant is:
b² - 4ac
That's it. Plug in your values, calculate, and check the sign. You immediately know what you're dealing with.
The Three Cases Explained
Your discriminant value falls into one of three categories. Each tells you something specific about your roots.
- Positive discriminant → Two real, distinct roots
- Zero discriminant → One real root (repeated)
- Negative discriminant → Two complex roots (conjugate pairs)
No guessing. No graphing. Just arithmetic.
Discriminant Roots Chart
| Discriminant Value | Root Type | Number of Roots | Graph Behavior |
|---|---|---|---|
| b² - 4ac > 0 | Real and unequal | 2 | Parabola crosses x-axis twice |
| b² - 4ac = 0 | Real and equal | 1 (double root) | Parabola touches x-axis at vertex |
| b² - 4ac < 0 | Complex conjugates | 0 real | Parabola stays entirely above or below x-axis |
How to Determine Root Type: Step by Step
Here's exactly what you do when facing a quadratic equation:
- Identify a, b, and c from your equation
- Calculate b² - 4ac
- Check the sign of your result
- Match to the chart above
That's the entire process. Three steps. No exceptions.
Working Examples
Example 1: Two Real Roots
Equation: 2x² + 5x - 3 = 0
a = 2, b = 5, c = -3
Discriminant = (5)² - 4(2)(-3) = 25 + 24 = 49
49 > 0, so you have two distinct real roots.
Example 2: One Repeated Root
Equation: x² - 6x + 9 = 0
a = 1, b = -6, c = 9
Discriminant = (-6)² - 4(1)(9) = 36 - 36 = 0
Zero discriminant means one root repeated twice. The root is x = 3.
Example 3: Complex Roots
Equation: x² + 2x + 5 = 0
a = 1, b = 2, c = 5
Discriminant = (2)² - 4(1)(5) = 4 - 20 = -16
Negative discriminant gives you complex conjugates: -1 ± 2i.
Why This Chart Matters
Most students waste time solving the entire quadratic formula only to discover they have complex roots. The discriminant check takes five seconds and tells you the answer before you touch the formula.
For exams, it also tells you how to sketch the graph. A positive discriminant means the parabola crosses the x-axis twice. Zero means it just touches. Negative means it never touches.
You can determine all of this without a single calculation involving the square root symbol.
Quick Reference
- Positive → 2 real roots
- Zero → 1 real root
- Negative → 2 complex roots
Memorize that. You won't need anything else.