Question 171214
{{{b^2-4ac}}} is called the discriminant of the quadratic equation, {{{x = (-b +- sqrt( b^2-4ac ))/(2a) }}}.  Note that this is the expression under the radical in the quadratic equation.


{{{DELTA}}} is the symbol for the discriminant, so {{{DELTA=b^2-4ac}}} 


If {{{DELTA=0}}} then there is a single root to the quadratic with multiplicity of 2 and that root is {{{x=(-b)/2a}}}.  The term multiplicity of 2 arises from the fact that every quadratic can be represented as two factors: {{{(x-r)(x-s)=0}}} where {{{r}}} and {{{s}}} can be either complex or real numbers.  In the case of {{{DELTA=0}}}, {{{r=s}}}.


If {{{DELTA>0}}} then there are two distinct real number roots.


If {{{DELTA<0}}} then there is a conjugate pair of complex roots of the form {{{alpha+-beta*i}}} where {{{i}}} is the imaginary number defined by {{{i^2=-1}}}