Discriminant of a 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 determinant is not generally calculated, its value has no interest.

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

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

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

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.