Some days ago I was solving a question which gave me a hard time. After doing some research I found out that it required the Discriminant of a Cubic Equation.
I looked up to the internet and I found out that the Discriminant of a Cubic Equation is $a^2b^2 + 18abc − 4b^3 − 4a^3c − 27c^2$.
So I was wondering if we have a general formula for finding the Discriminant an Equation of any degree.
I have been taught the discriminant of a Quadratic Equation where we prove it by Completing the Square method so I was wondering if there was any method to find/prove if for Cubic Equations.
So, the 2 main Question I want to ask are,
How can we find/prove the Discriminant of a Cubic Equation?
And,
Can we find the Discriminant of any equation of any degree?
Any help would be appreciated.
Best Answer
I think your formula for the discriminant of a cubic equation is wrong. This formula is independent of $d$. (Assuming that you considered the cubic equation as $ax^3+bx^2+cx+d$). The correct formula for the discriminant is given by $D = b^2c^2-4ac^3-4b^3d-27a^2d^2+18abcd.$ Refer https://brilliant.org/wiki/cubic-discriminant/ for detailed proof.
Also, Have a look at a simiar question asked here: Proof of the cubic discriminant
A general formula for finding the discriminant of an equation of any degree $(P(x)=a_nx^n+a_{n−1}x^{n−1}+\cdots+a_1x+a_0$ having roots $x_1,x_2,\ldots,x_n)$ is given by:
$D = a_n^{2n-2} \prod_{1 \leq i < j \leq n}{(x_i-x_j)^2}.$
Moreover, you can refer to Galois Theory by Ian Stewart. The book has a nice treatment on the solution of polynomial equation in section $1.4$ of chapter $1$.