2
$\begingroup$

Quadratic equation looks like that: $$ax^2+bx+c=0$$ where $a\ne 0$. We can say something about roots when We compute a discriminant $$\Delta=b^2-4ac$$ When $\Delta>0$ then We have two real roots, when $\Delta<0$, then two imaginary roots. Can We say enything about amount of thise equations in finite set of coefficients? E.g. when $a,b,c\in\{1,2,...,n\}$ then We have $n^3$ different equations. I prepared graphs when $a,b,c\in\{1,2,...,i\}$ and $i$ is on horizontal axis. On vertical axis there is amount of particular equations.Amount of quadratic equations depend on coefficients

At the first I thought that this is because of asymmetric shape of formula for $\Delta$. $ac$ is multipled by $4$. When I divide amount of $\Delta<0$ equantions by $4$, then I got: Amount of quadratic equations depend on coefficients

Is there any theorem or results about that? Maybe in countinuous sets We can say something about ratios?

$\endgroup$
0
$\begingroup$

If $N$ is the number of equations with $\Delta < 0$, the number of equations with $\Delta > 0$ seems to be much closer to $N/3$ than to $N/4$. That is, for every $4$ equations, on average, we typically get one equation where $\Delta > 0$ and the other three have $\Delta < 0$. You could work out the asymptotic behavior of this ratio by considering the probability that $X^2 > 4YZ$ where $X$, $Y$, and $Z$ are iid uniform continuous random variables.

So yes, we can say something about ratios, and very likely someone already has do so.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.