〜 ★ dCode presents ★ 〜

# Discriminant of a Polynomial

Results
Tool to compute the discriminant of a polynomial. A discriminant of a polynomial is an expression giving information about the nature of the roots of the polynomial.
Summary

## Discriminant Calculator

### How to calculate a discriminant?

For a quadratic polynomial $ax^2+bx+c$, the discriminant named delta $\Delta$ is calculated with the formula:

$$\Delta = b^2-4ac$$

The fact of knowing the value of the discriminant then solves the equation more easily through formulas (using this discriminant).

Example: The equation $2x^2+3x+1 = 0$ of type $ax^2+bx+c$ (with $a = 2$, $b = 3$ et $c = 1$) has for discriminant $\Delta = b^2-4ac = 3^2-4*2*1 = 1$

For a cubic polynomialof the form $$ax^3+bx^2+cx+d$$ the discriminant formula is

$$\Delta = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd$$

For a polynomial of degree 1 or 0 the determinant is not generally calculated, its value has no interest.

### How to find the roots of a polynomial with the determinant?

For a quadratic polynomial of type $ax^2+bx+c = 0$

If the discriminant is positive (strictly), the equation has two solutions:

$$x_1 = \frac {-b + \sqrt \Delta}{2a} \\ x_2 = \frac {-b - \sqrt \Delta}{2a}$$

Example: The equation $2x^2+3x+1 = 0$ has for discriminant $\Delta = 1$, so solutions are $x_1 = -1/2$ and $x_2 = -1$

If the discriminant is zero, the equation has a double root:

$$x_1=x_2 = -\frac b{2a}$$

If the discriminant is negative (strictly), the equation has 2 complex conjugate solutions:

$$\delta^2 = \Delta$$

$$x_1 = \frac {-b + \delta}{2a} \\ x_2 = \frac {-b - \delta}{2a}$$

For equations of higher degrees, the calculations are much more complicated, but knowledge of the determinants is important.

## Source code

dCode retains ownership of the source code of the script Discriminant of a Polynomial online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Discriminant of a Polynomial script for offline use on PC, iPhone or Android, ask for price quote on contact page !