Is there a situation such that the 'magnitude' of discriminant is important?

Question

For example, we know x^2-x+1 = 0 has discriminant -3, its sign is minus, so it has two different non-real roots.

The sign of discriminant is also useful for inspecting 'root profile' of higher(cubic, quartic) degree polynomials. Whether it has any multiple roots or not, whether it has non-real root..
Also I heard that the sign of discriminant plays a role to judge whether the polynomial can be solved purely algebraically.

My question is : is there any

meaningful or useful aspect of the magnitude(=absolute value) of discriminant

? If not.. then personally I want to name 'sign of discriminant' to 'root indicator', and will not use the term 'discriminant'.

The comments/answers so far are about quadratic equation. Hope there is something for cubic,quartic,.. or general n-th degree equation.

Answer 1

If $r_1$ and $r_2$ are the roots of $x^2+ax+b$ and if $\Delta$ is its discriminant, then $|\Delta|=|r_1-r_2|^2$. So, you can use $|\Delta|$ to find the distance between the roots.

Answer 2

If the given quadratic is $ax^2+bx+c$ with $\Delta=b^2-4ac$, then

$ax^2+bx+c=\frac{1}{4a}\left( 2ax +b \right)^2 - \frac{\Delta}{4a}$, which means the minimum or maximum value of the quadratic ($- \frac{\Delta}{4a}$) is also determined by the magnitude $\Delta$.

Answer 3

I will add one for cubic equation :

Let $$F(x) = ax^3 + bx^2 + cx +d$$ where $a,b,c,d$ are integers with $a$ nonzero.

The discriminant of F is

$$ Disc[F] = b^2 c^2 - 4 a c^3 - 4 b^3 d + 18 a b c d - 27 a^2 d^2$$

Following fact is noteworthy :

There is a root of $F[x]=0$, with a form $$p+ \sqrt[3]q + \sqrt[3]r$$

where $p,q,r$ are rational numbers, if $Disc[F]$ is a from of $-3k^2$ where $k$ is an integer.

Such example relies on the whole value(I mean both sign and magnitude) of discriminant.