Question 113339
By D, I presume you mean the discriminant {{{b^2-4ac}}} part of the quadratic formula.


When D < 0, the solution to the equation will be a conjugate pair of complex number roots.  This is because {{{sqrt(-1)}}} is not defined in the real number system so we have to use the imaginary number i where {{{i^2=-1}}} to express the roots of the equation.  This conjugate pair of complex numbers is always in the form of {{{a+-bi}}}, though it is possible for the real number part of the complex number to be zero, as in the solution to {{{x^2+5=0}}}  (The solutions are {{{i*sqrt(5)}}} and {{{-i*sqrt(5)}}} which are equivalent to {{{0+i*sqrt(5)}}} and {{{0-i*sqrt(5)}}}.  Considering the function {{{f(x)=ax^2+bx+c}}}, if {{{b^2-4ac<0}}}, then the graph of the function will not intersect the x-axis.


An example of such an equation is {{{x^2+2x+5}}}.  Here, the discriminant is {{{4-20=-16}}}, and the solution set would contain a conjugate pair of complex roots given by:


{{{x=(-2+-i*sqrt(16))/2}}}
{{{x=-1+2i}}} or {{{x=-1-2i}}}


{{{graph(400,400,-5,5,-2,10,x^2+2x+5)}}}


Notice that the graph does NOT intersect the x-axis.


Hope this helps.
John

P.S. Super-Double-Plus Extra Credit.  Now that we know that there are ALWAYS two roots to a second degree or quadratic equation, how many roots would you say a third degree or cubic equation ALWAYS has?