Discriminant of a Polynomial

A discriminant of a polynomial is an expression giving information about the number and the nature of the roots of the polynomial.

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).

For a cubic polynomial of 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 discriminant is not generally calculated, its value has no interest, it is nevertheless sometimes defined at the value $ 1 $.

The discriminant acts as an indicator of whether the polynomial has real, distinct or imaginary roots and their number.

For a polynomial of degree 2 if the discriminant is equal to $ 0 $, then the polynomial has a unique (double) root. If the discriminant is positive, then it has 2 real roots, and if it is negative, it has 2 complex and conjugate roots.

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

If the discriminant is positive (strictly), the equation has two solutions x1 and x2:

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

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 (degree 3 or 4 or more), knowledge of the discriminant allows us to know the number of roots, however there is no formula allowing them to be found from the discriminant.