First subtract #n^2-4n# from both sides to get:
#4n^2+10n+7 = 0#
This is of the form #an^2+bn+c = 0#, with #a=4#, #b=10# and #c=7#
The discriminant is given by the formula:
#Delta = b^2-4ac = 10^2-(4xx4xx7) = 100-112 = -12#
Since #Delta < 0# the quadratic has no real solutions. It has two distinct complex roots.
In general, the possible cases are:
#Delta = 0# : Means the quadratic has one repeated real root. If the coefficients of the quadratic are rational, that repeated root is also rational.
#Delta > 0# : Means that the quadratic has two distinct real roots. If #Delta# is also a perfect square and the coefficients of the quadratic are rational, then those roots are also rational.
#Delta < 0# : Means that the quadratic has no real roots. It has two distinct complex roots which are complex conjugates of one another.
