Discriminant

Calculator to determine the number and type of solutions of quadratic equations

Discriminant Calculator

What is calculated?

The discriminant D = b² - 4ac of a quadratic equation ax² + bx + c = 0 determines the number and type of solutions. It is the expression under the square root in the quadratic formula.

Coefficients of the quadratic equation
ax² + bx + c = 0
a ≠ 0 required
Linear term
Absolute term
Result
Discriminant D = b² - 4ac
The discriminant determines the number of real solutions of the quadratic equation

Discriminant Info

Interpretation

Discriminant D = b² - 4ac:

D > 0: Two real solutions D = 0: One real solution D < 0: No real solutions

Geometrically: The discriminant indicates how many times the parabola intersects the x-axis.

Quick examples
x² - 4 = 0:
D = 0² - 4(1)(-4) = 16 > 0
→ Two solutions: x = ±2
x² - 2x + 1 = 0:
D = (-2)² - 4(1)(1) = 0
→ One solution: x = 1
x² + 1 = 0:
D = 0² - 4(1)(1) = -4 < 0
→ No real solutions
Related topics

→ Quadratic Equations
→ Quadratic Formula
→ Find Roots

Formulas and relations

Discriminant
\[D = b^2 - 4ac\] Basic formula
Quadratic formula
\[x_{1,2} = \frac{-b \pm \sqrt{D}}{2a}\] With D = b² - 4ac
Two real solutions (D > 0)
\[x_1 = \frac{-b + \sqrt{D}}{2a}\] \[x_2 = \frac{-b - \sqrt{D}}{2a}\]
One real solution (D = 0)
\[x = \frac{-b}{2a}\] Double root
Complex solutions (D < 0)
\[x_{1,2} = \frac{-b \pm i\sqrt{|D|}}{2a}\] With i = √(-1)
Vieta's formulas
\[x_1 + x_2 = -\frac{b}{a}\] \[x_1 \cdot x_2 = \frac{c}{a}\]
Vertex
\[S = \left(-\frac{b}{2a}, -\frac{D}{4a}\right)\] Parabola vertex
Distance between roots
\[|x_1 - x_2| = \frac{\sqrt{D}}{|a|}\] If D > 0

Detailed worked example

Example: -2x² + 3x + 5 = 0

Given:

  • a = -2 (coefficient of x²)
  • b = 3 (coefficient of x)
  • c = 5 (constant term)

Step 1 - Compute discriminant:

\[D = b^2 - 4ac\] \[D = 3^2 - 4(-2)(5)\] \[D = 9 + 40 = 49\]

Result: D = 49 > 0

→ Two distinct real solutions exist!

Step 2 - Compute solutions:

\[x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{2(-2)} = \frac{-3 \pm 7}{-4}\] \[x_1 = \frac{-3 + 7}{-4} = \frac{4}{-4} = -1\] \[x_2 = \frac{-3 - 7}{-4} = \frac{-10}{-4} = 2.5\]

Geometric interpretation

Parabola and x-axis intersections
D > 0: Two intersections

Parabola intersects x-axis twice

D = 0: One tangent point

Parabola is tangent to the x-axis once

D < 0: No intersection

Parabola lies above the x-axis

Practical applications

Physics
  • Projectile parabola calculations
  • Vibration analysis
  • Optimization problems
  • Determine trajectories
Economics
  • Profit and cost functions
  • Break-even analyses
  • Market equilibrium
  • Optimal production quantities
Engineering
  • Parabolic antenna design
  • Bridge construction
  • Signal processing
  • Control system design

Historical context

Development of the theory
  • Babylonians (~2000 BC): Early quadratic equations
  • Al-Khwarizmi (~830): Systematic solutions
  • Vieta (1540-1603): Symbolic algebra
  • Modern form: 17th/18th century
Mathematical significance
  • Fundamental concept of algebra
  • Generalization to higher polynomials
  • Galois theory and solvability
  • Complex numbers originated from it
Summary

The discriminant is a central concept in algebra that characterizes the solvability of quadratic equations. It connects algebraic and geometric aspects and has broad applications in science and engineering. Its development was crucial for the emergence of modern algebra.

Mathematical Equations

Associative Property  •  Linear equation  •  Quadratic equation  •  Cubic equation  •  Discriminant  •  Gauss elimination method  •