Discriminant Calculator
Calculate polynomial discriminants and analyze root types for quadratic, cubic, and higher-degree equations
Polynomial Discriminant Calculator
Polynomial:
Discriminant Results
Sign: Zero
Root Type: Multiple Root
Analysis
Description: The polynomial has one double root (repeated root)
Root Count: 1 repeated root
Geometric: The parabola touches the x-axis at one point
Discriminant Formula
Example Problems
Example 1: Quadratic Discriminant
Polynomial: x² - 5x + 6
Coefficients: a = 1, b = -5, c = 6
Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
Result: Δ > 0, so there are 2 distinct real roots
Example 2: Perfect Square
Polynomial: x² - 4x + 4
Coefficients: a = 1, b = -4, c = 4
Discriminant: Δ = (-4)² - 4(1)(4) = 16 - 16 = 0
Result: Δ = 0, so there is 1 repeated root (x = 2)
Discriminant Rules
Multiple distinct real roots (for quadratic: 2 real roots)
At least one repeated root (multiple roots)
Complex roots (for quadratic: 2 complex conjugates)
Polynomial Degrees
Quadratic (Degree 2)
ax² + bx + c
Cubic (Degree 3)
ax³ + bx² + cx + d
Quartic (Degree 4)
ax⁴ + bx³ + cx² + dx + e
Quintic (Degree 5)
ax⁵ + bx⁴ + cx³ + dx² + ex + f
Understanding Discriminants
What is a Discriminant?
The discriminant is a mathematical expression calculated from the coefficients of a polynomial that reveals information about the nature of its roots without actually solving the equation. It tells us whether roots are real or complex, and whether any roots are repeated.
Why is it Important?
- •Quickly determines root types without solving
- •Helps choose appropriate solution methods
- •Predicts graphical behavior of polynomials
- •Essential in algebraic analysis
Quadratic Examples
Two Real Roots
x² - 3x + 2: Δ = 9 - 8 = 1 > 0
Roots: x = 1, x = 2
One Repeated Root
x² - 4x + 4: Δ = 16 - 16 = 0
Root: x = 2 (double)
Complex Roots
x² + x + 1: Δ = 1 - 4 = -3 < 0
Roots: (-1 ± i√3)/2
Higher Degree Polynomials
Cubic
5-term discriminant formula determines if all three roots are real
Quartic
16-term formula with complex root pattern analysis
Quintic
59-term formula requiring advanced computation