Loading the page...
Preparing tools and content for you. This usually takes a second.
Preparing tools and content for you. This usually takes a second.
Fetching calculator categories and tools for this section.
Free discriminant calculator for quadratic equations. Calculate discriminant, analyze root types, and determine parabola intersection behavior. Our calculator uses the discriminant formula Δ = b² - 4ac to determine the nature and number of roots for any quadratic equation.
Last updated: February 2, 2026
Need a custom algebra calculator for your educational platform? Get a Quote
Quadratic Equation
Coefficient of x²
Coefficient of x
Constant term
Discriminant (Δ = b² - 4ac)
1
Root Type:
Two distinct real roots
Number of Roots:
2
Roots:
x₍1₎ = 3
x₍2₎ = 2
Graph Behavior:
Parabola intersects x-axis at two distinct points
Analysis:
The quadratic equation has two distinct real solutions
Factorization:
Can be factored into linear terms
Discriminant Rules:
Formula
Δ = b² - 4ac
Determines root types for ax² + bx + c = 0
Classifications
Real, Repeated, Complex
Analyzes Δ > 0, Δ = 0, and Δ < 0 cases
Intersections
0, 1, or 2 points
Visual representation of quadratic graph behavior
Factorability
Perfect Square Test
Checks if discriminant is a perfect square
Complex form
a ± bi
Handles imaginary solutions when Δ < 0
Full formula
x = (-b ± √Δ) / 2a
Discriminant as part of complete solution process
For quadratic equation x² - 5x + 6 = 0:
Discriminant (Δ)
1
Root Type
Two distinct real roots
Our discriminant calculator analyzes quadratic equations using the discriminant formula Δ = b² - 4ac to determine root types and parabola behavior. The calculation applies fundamental algebraic principles to classify solutions and provide insights into quadratic function properties.
Δ = b² - 4acΔ > 0
Two distinct real roots
Parabola crosses x-axis twice
Δ = 0
One repeated real root
Parabola touches x-axis once
Δ < 0
Two complex roots
Parabola doesn't touch x-axis
Shows three cases of parabola behavior based on discriminant value
The discriminant is derived from the quadratic formula and reveals the nature of solutions before actually computing them. It's the expression under the square root in the quadratic formula, making it a powerful tool for analysis without full calculation.
Need help with other algebra calculations? Check out our quadratic formula calculator and partial fractions calculator.
Get Custom Calculator for Your PlatformResult: Δ = 25 > 0 (Two distinct real roots)
Since 25 is a perfect square, the roots are rational: x = 3 and x = 1/2
x² - 6x + 9 = 0
Δ = 0 (One repeated root: x = 3)
x² + x + 1 = 0
Δ = -3 < 0 (Two complex roots)
Share it with others who need help with quadratic equation analysis
Suggested hashtags: #Algebra #Discriminant #Quadratic #Mathematics #Calculator