Question 155038
For this discussion, we'll use the discriminant.
For the general quadratic equation, 
{{{ax^2+bx+c=0}}}
the discriminant is 
{{{D=b^2-4ac}}}
.
.
If {{{D>0}}} then you have two distinct real roots.
If {{{D=0}}}, you have a double root, one real root occurring twice 
If {{{D<0}}}, you have two complex roots, that are complex conjugates.

{{{x^2+px+1=0}}}
{{{D=p^2-4(1)(1)}}}
{{{D=p^2-4}}}
For no real (complex) solutions,
{{{D<0}}}
{{{p^2-4<0}}}
{{{p^2<4}}}
Valid interval: ({{{-2<p<2}}})
.
.
.
.
For two real solutions,
{{{D>=0}}}
{{{p^2>=0}}}
{{{p>=2}}} and {{{p<=-2}}}
Valid interval:({{{-infinity}}},{{{-2}}}]U[{{{2}}},{{{infinity}}})