Question 70739
The roots, or x-intercepts, are found by the formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
rewrite this as
{{{x = (-b / 2a) +- sqrt( b^2-4*a*c ) / (2*a) }}}
The {{{b^2 - 4ac}}} under the square root sign
determines the nature of the x-intercepts (roots).
If {{{b^2 = 4ac}}}, the whole 2nd term is zero, and
there is 1 x-intercept at {{{-b/(2a)}}}. All this 
means is that the vertex of the parabola just kisses
the x-axis.
If {{{b^2 > 4ac}}}, There are 2 x-intercepts at these locations:
{{{x = (-b / 2a) + sqrt( b^2-4*a*c ) / (2*a) }}}
and
{{{x = (-b / 2a) - sqrt( b^2-4*a*c ) / (2*a) }}}
If {{{b^2 < 4ac}}}, There are a pair of imaginary roots
(involving {{{sqrt(-1)}}})
All that means is that the parabola never touches the x-axis