Question 909131
The second part of the description seems to mean, {{{sqrt(m/n)}}} and you want m-n; and that distance between x values of the the intersection points is {{{sqrt(m/n)}}}. 


{{{3=4x^2+x-1}}}
{{{4x^2+x-4=0}}}
{{{x=(-1+- sqrt(1+16))/8}}}
The x values, {{{highlight(x=(-1+-sqrt(17))/8)}}}


Distance Between Those x values:
{{{((-1-sqrt(17))-(-1+sqrt(17)))/8}}}
{{{(-2sqrt(17))/8}}}
{{{-sqrt(17)/4}}}


I am stopping here because of the lack of clarity in part of your description, but you may be able to understand what you need from the steps presented.