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Discriminant Calculator

Select the polynomial type and write down its coefficients. The discriminant calculator determines the discriminant of it with detailed calculations displayed.

Polynomial's degree:

Enter a,b,c in ax² + bx + c = 0

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The discriminant calculator is a powerful tool that evaluates the discriminant of quadratic, cubic, quartic, and higher-degree polynomials. It helps determine how many roots an equation has and whether those roots are real, complex, equal, rational, or irrational.

What is a Discriminant?

In algebra, the discriminant is a mathematical expression calculated from the coefficients of a polynomial. It is denoted by the symbol (Δ) and provides key insights into the characteristics of the equation’s roots without actually solving it.

Understanding the Nature of Roots

For Quadratic Equations

If (Δ > 0) and not a perfect square → two distinct real irrational roots.
If (Δ = 0) → one repeated real root.
If (Δ < 0) → two complex (non-real) roots.
If (Δ) is a perfect square → two distinct real rational roots.

For Cubic Equations

If (Δ > 0) → three distinct real roots.
If (Δ < 0) → one real root and two complex conjugate roots.
If (Δ = 0) → at least two roots are equal.

For Quartic Equations

If (Δ > 0) → four real roots.
If (Δ < 0) → two real roots and two complex conjugates.
If (Δ = 0) → repeated roots exist. Possible cases include:

Four real roots with one double root.
Two separate double real roots.
One triple root and one single root.
A single root repeated four times.
A double real root with two complex roots.
Two pairs of repeated complex roots.

Total Number of Roots by Degree

Quadratic polynomial → 2 roots
Cubic polynomial → 3 roots
Quartic polynomial → 4 roots

This calculator instantly evaluates the discriminant and explains the number and type of roots for polynomials up to the fourth degree.

Relationship Between Discriminant and Parabolas

(Δ > 0) → parabola crosses the x-axis at two points.
(Δ < 0) → parabola does not intersect the x-axis.
(Δ = 0) → parabola touches the x-axis at exactly one point.

General Formula for the Discriminant

For a polynomial written as p(x) = a_nxⁿ + ... + a₁x + a₀ with roots r₁, r₂, ..., r_n:

D(p) = a_n^2n-2 ∏ (r_i - r_j)², for i < j

The discriminant is a homogeneous expression in the coefficients.
Its value does not depend on the order of the roots.

Quadratic Formula

ax² + bx + c = 0

(Δ) = b² - 4ac

Cubic Formula

ax³ + bx² + cx + d = 0

(Δ) = b²c² - 4ac³ - 4b³d - 27a²d² + 18abcd

Quartic Formula

ax⁴ + bx³ + cx² + dx + e = 0

(Δ) = 256a³e³ - 192a²bde² - 128a²c²e² + 144a²cd²e - 27a²d⁴ + 144ab²ce² - 6ab²d²e - 80abc²de + 18abcd³ + 16ac⁴e - 4ac³d² - 27b⁴e² + 18b³cde - 4b³d³ - 4b²c³e + b²c²d²

Complexity of Higher-Degree Discriminants

Cubic polynomial → contains 5 terms
Quartic polynomial → contains 16 terms
Quintic polynomial → contains 59 terms
Sextic polynomial → contains 246 terms
Septic polynomial → contains 1103 terms

How the Discriminant Calculator Operates

Our calculator evaluates the discriminant automatically and displays a detailed breakdown of each calculation step.

Required Inputs

Choose the polynomial degree.
Provide the coefficients in the appropriate fields.
Press the “Calculate” button.

Generated Results

Computed discriminant value.
Explanation of the root type.
Complete step-by-step solution process.

Worked Examples

Example 1: Quadratic

Given equation: 4x² - 5x + 1 = 0

a = 4, b = -5, c = 1

(Δ) = (-5)² - 4(4)(1) = 25 - 16 = 9

Example 2: Cubic

Given equation: 3x³ + x² - 4x + 2 = 0

a = 3, b = 1, c = -4, d = 2

After applying the formula, (Δ) = -4284

Example 3: Quartic

Given equation: x⁴ + 2x³ + x² + 5x + 6 = 0

a = 1, b = 2, c = 1, d = 5, e = 6

After evaluation, (Δ) = 13824

References

From educational math resources discussing polynomial discriminants and root behavior.

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